Multiphase Flow Dynamics. Theory and Numerics [Monifaasivirtausten Dynamiikka

VTT PUBLICATIONS 722

Kai Hiltunen, Ari Jäsberg, Sirpa Kallio, Hannu Karema, Markku Kataja, Antti Koponen, Mikko Manninen & Veikko Taivassalo

Multiphase Flow Dynamics

Theory and Numerics

VTT PUBLICATIONS 722

Multiphase Flow Dynamics Theory and Numerics

Kai Hiltunen, Ari Jäsberg, Sirpa Kallio, Hannu Karema, Markku Kataja, Antti Koponen, Mikko Manninen & Veikko Taivassalo

ISBN 978-951-38-7365-3 (soft back ed.) ISSN 1235-0621 (soft back ed.) ISBN 978-951-38-7366-0 (URL: http://www.vtt.fi/publications/index.jsp) ISSN 1455-0849 (URL: http://www.vtt.fi/publications/index.jsp) Copyright © VTT 2009

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Edita Prima Oy, Helsinki 2009

Kai Hiltunen, Ari Jäsberg, Sirpa Kallio, Hannu Karema, Markku Kataja, Antti Koponen, Mikko Manninen & Veikko Taivassalo. Multiphase Flow Dynamics. Theory and Numerics [Monifaasivirtausten dynamiikka. Teoriaa ja numeriikkaa]. Espoo 2009. VTT Publications 722. 113 p. + app. 4 p.

Keywords multiphase flows, volume averaging, ensemble averaging, mixture models, multifluid finite volume method, multifluid finite element method, particle tracking, the lattice- BGK model

Abstract

The purpose of this work is to review the present status of both theoretical and numerical research of multiphase flow dynamics and to make the results of that fundamental research more readily available for students and for those working with practical problems involving multiphase flow. Flows that appear in many of the common industrial processes are intrinsically multiphase flows – e.g. flows of gas-particle suspensions, liquid-particle suspensions, and liquid-fiber suspensions, as well as bubbly flows, liquid-liquid flows, and the flow through porous medium. In the first part of this publication we give a comprehensive review of the theory of multiphase flows accounting for several alternative approaches. The second part is devoted to numerical methods for solving multiphase flow equations.

3 Kai Hiltunen, Ari Jäsberg, Sirpa Kallio, Hannu Karema, Markku Kataja, Antti Koponen, Mikko Manninen & Veikko Taivassalo. Multiphase Flow Dynamics. Theory and Numerics [Monifaasivirtausten dynamiikka. Teoriaa ja numeriikkaa]. Espoo 2009. VTT Publications 722. 113 s. + liitt. 4 s.

Avainsanat multiphase flows, volume averaging, ensemble averaging, mixture models, multifluid finite volume method, multifluid finite element method, particle tracking, the lattice- BGK model

Tiivistelmä

Työssä tarkastellaan monifaasivirtausten teoreettisen ja numeerisen tutkimuksen nykytilaa, ja muodostetaan tuon perustutkimuksen tuloksista selkeä kokonaisuus opiskelijoiden ja käytännön virtausongelmien kanssa työskentelevien käyttöön. Monissa teollisissa prosesseissa esiintyvät virtaukset ovat olennaisesti monifaasivirtauksia – esimerkiksi kaasu-partikkeli-, neste-partikkeli- ja neste-kuitu- suspensioiden virtaukset, sekä kuplavirtaukset, neste-neste-virtaukset ja virtaus huokoisen aineen läpi. Julkaisun ensimmäisessä osassa tarkastellaan kattavasti monifaasivirtausten teoriaa ja esitetään useita vaihtoehtoisia lähestymistapoja. Toisessa osassa käydään läpi monifaasivirtauksia kuvaavien yhtälöiden numeeri- sia ratkaisumenetelmiä.

4 Preface

This monograph was originally compiled within the project ”Dynamics of Multiphase Flows” which was a part of the Finnish national Computational Fluid Dynamics Technology Programme 1995–1999. The purpose of this work is to review the present status of both theoretical and numerical research of multiphase flow dynamics and to make the results of that fundamental research more readily available for students and for those working with practical problems involving multiphase flow. Indeed, flows that appear in many of the common industrial processes are intrinsically multiphase flows. For example, gas-particle suspensions or liquid-particle suspensions appear in combustion processes, pneumatic conveyors, separators and in numerous processes within chemical industry, while flows of liquid-fiber suspensions are essential in paper and pulp industry. Bubbly flows may be found in evaporators, cooling systems and cavitation processes, while liquid-liquid flows frequently appear in oil extraction. A specific category of multiphase flows is the flow through porous medium which is important in filtration and precipitation processes and especially in numerous geophysical applications within civil and petroleum engineering. The advanced technology associated with these flows has great economical value. Nevertheless, our basic knowledge and understanding of these processes is often quite limited as, in general, is our capability of solving these flows. In this respect, the condition within multiphase flow problems is very much different from the conventional single phase flows. At present, relatively reliable models exist and versatile commercial computer programs are available and capable of solving even large scale industrial problems of single phase flows. Advanced commercial computer codes now include features which also facilitate numerical simulation of multiphase flows. In most cases, however, a realistic numerical solution of practical multiphase flows requires, not only a powerful computer and an effective code, but deep understanding of the physical content and of the nature of the equations that are being solved as well as of the underlying dynamics of the microscopic processes that govern the observed behaviour of the flow. In the first part of this monograph we give a comprehensive review of the theory of multiphase flows accounting for several alternative approaches. We also give general quidelines for solving the ’closure problem’, which involves,

5 Preface e.g., characterising the interactions between different phases and thereby deriving the final closed set of equations for the particular multiphase flow under consideration. The second part is devoted to numerical methods for solving those equations.

6 Contents

Abstract 3 Tiivistelmä 4 Preface 5

1 Equations of multiphase ﬂow 9 1.1Introduction...... 9 1.2Volumeaveraging ...... 12 1.2.1 Equations...... 12 1.2.2 Constitutiverelations...... 17 1.3Ensembleaveraging...... 23 1.4Mixturemodels...... 27 1.5Particletrackingmodels...... 32 1.5.1 Equationofmotionforasingleparticle...... 33 1.5.2 Particledispersion...... 36 1.6Practicalclosureapproaches...... 45 1.6.1 Diluteliquid-particlesuspension...... 45 1.6.2 Flowinaporousmedium...... 48 1.6.3 Densegas-solidsuspensions...... 51 1.6.4 Constitutive equations for the mixture model . . . . . 55 1.6.5 Dispersionmodels...... 58

2Numericalmethods 64 2.1Introduction...... 64 2.2MultiﬂuidFiniteVolumeMethod...... 66 2.2.1 Generalcoordinates...... 67 2.2.2 Discretizationofthebalanceequations...... 69 2.2.3 Rhie-Chowalgorithm...... 75 2.2.4 Inter-phasecouplingalgorithms...... 77 2.2.5 Solutionofvolumefractionequations...... 79 2.2.6 Pressure-velocitycoupling...... 80 2.2.7 Solutionalgorithm...... 83 2.3MultiﬂuidFiniteElementMethod...... 84 2.3.1 StabilizedFiniteElementMethod...... 86

7 2.3.2 Integrationandisoparametricmapping...... 90 2.3.3 Solutionofthediscretizedsystem...... 91 2.4Particletracking...... 93 2.4.1 Solution of the system of equation of motion . . . . . 94 2.4.2 Solutionofparticletrajectories...... 95 2.4.3 Sourcetermcalculation...... 96 2.4.4 Boundary condition ...... 96 2.5Mesoscopicsimulationmethods ...... 96 2.5.1 Thelattice-BGKmodel...... 98 2.5.2 Boundary conditions ...... 99 2.5.3 Liquid-particlesuspensions...... 100 2.5.4 Applicability of mesoscopic methods ...... 101

Bibliography 104

Appendix 1

8 1. Equations of multiphase ﬂow

1.1 Introduction

A multiphase fluid is composed of two or more distinct components or ’phases’ which themselves may be fluids or solids, and has the characteristic properties of a fluid. Within the dicipline of multiphase flow dynamics the present status is quite different from that of the single phase flows. The theoretical background of the single phase flows is well established (the crux of the theory being the Navier-Stokes equation) and apparently the only outstanding practical problem that still remains unsolved is turbulence, or perhaps more generally, problems associated with flow stability. While it is rather straightforward to derive the equations of the conservation of mass, momentum and energy for an arbitrary mixture, no general counterpart of the Navier-Stokes equation for multiphase flows have been found. Using a proper averaging procedure it is however quite possible to derive a set of ”equations of multiphase flows” which in principle correctly describes the dynamics of any multiphase system and is subject only to very general assumptions (see section 1.2 below). The drawback is that this set of equations invariably includes more unknown variables than independent equations, and can thus not be solved. In order to close this set of equations, additional system dependent costitutive relations and material laws are needed. Considering the many forms of industrial multiphase flows, such as flow in a fluidized bed, bubbly flow in nuclear reactors, gas-particle flow in combustion reactors and fiber suspension flows within pulp and paper industry, it seems virtually impossible to infer constitutive laws that would correctly describe interactions and material properties of the various phases involved, and that would be common even for these few systems. Furthermore, even in a laminar flow of, e.g., liquid-particle suspensions, the presence of particles induces fluctuating motion of both particles and fluid. Analogously to the Reynolds stresses that arise from time averaging the turbulent motion of a single phase fluid, averaging over this ”pseudo-turbulent” motion in multiphase systems leads to additional correlation terms that are unknown

9 1. Equations of multiphase flow a priori. For genuinely turbulent multiphase flows, the dynamics of the turbulence and the interaction between various phases are problems that presumably will elude general and practical solution for decades to come. A direct consequence of the complexity and diversity of these flows is that the dicipline of multiphase fluid dynamics is and may long remain a prominently experimental branch of fluid mechanics. Preliminary small scale model testing followed by a trial and error stage with the full scale system is still the only conceivable solution for many practical engineering problems involving multiphase flows. Inferring the necessary constitutive relations from measured data and verifying the final results are of vital importance also within those approaches for which theoretical modeling and subsequent numerical solution is considered feasible. In general, a multiphase fluid may be a relatively homogeneous mixture of its components, or it may be manifestly inhomogeneous in macroscopic scales. While much of the general flow dynamics covered by the present monograph can be applied to both types of multiphase fluids, we shall mainly ignore here macroscopically inhomogeneous flows such as stratified flow and plug flow of liquid and gas in a partially filled tube. In what follows we thus restrict ourselves to flows of macroscopically homogeneous multiphase fluids. In modeling such flows, several alternative approaches can be taken. Perhaps the most frequently used method is to treat the multicomponent mixture, e.g., a liquid-particle suspension, effectively as a single fluid with rheological properties that may depend on local particle concentration. This approach may be used in cases where the velocities of various phases are nearly equal and when the effect of the interactions between the phases can be adequately described by means of rheological variables such as viscosity. The advantage of these ’homogeneous models’ is that numerical solution may be attempted utilizing conventional single fluid algorithms and effective commercial programs. In some cases the method can be improved by adding a separate particle tracking feature or an additional particle transport mechanism superposed on the mean flow. Although various traditional methods based on a single fluid approach may be sufficient for predicting gross features of certain special cases of also multiphase flows, it has become increasingly clear, that in numerous cases of practical interest, an adequate description requires recognition of the underlying multiphase character of the system. Genuine models for multiphase flows have been developed mainly following two different approaches. Within the ’Eulerian approach’ all phases are treated formally as fluids which obey normal one phase equations of motion in the unobservable ’mesoscopic’ level (e.g, in the size scale of suspended particles) — with appropriate boundary conditions specified at phase boundaries. The macroscopic flow equations are derived from these mesoscopic equations using an averaging procedure of some kind. This averaging procedure can be carried out in several alternative ways such as time averaging

10 1. Equations of multiphase ﬂow

[Ish75, Dre83], volume averaging [Ish75, Dre83, Soo90, Dre71, DS71, Nig79] and ensemble averaging [Ish75, Dre83, Buy71, Hwa89, JL90]. Various combi- nations of these basic methods can also been considered [Ish75]. Irrespective of the method used, the averaging procedure leads to equations of the same generic form, namely the form of the original phasial equations with a few extra terms. These extra terms include the interactions (change of mass, momentum etc. ) at phase boundaries and terms analogous to the ordinary Reynold’s stresses in the turbulent single phase flow equations. Each averaging procedure may however provide a slightly different view in the physical interpretation of these additional terms and, consequently, may suggest different approach for solving the closure problem that is invariably associated with the solution of these equations. The manner in which the various possible interaction mechanisms are naturally divided between these additional terms, may also depend on the averaging procedure being used. The advantage of the Eulerian method is its generality: in principle it can be applied to any multiphase system, irrespective of the number and nature of the phases. A drawback of the straightforward Eulerian approach is that it often leads to a very complicated set of flow equations and closure relations. In some cases, however, it is possible to use a simplified formulation of the full Eulerian approach, namely ’mixture model’ (or ’algebraic slip model’). The mixture model may be applicable, e.g., for a relatively homogeneous suspension of one or more species of dispersed phase that closely follow the motion of the continuous carrier fluid. For such a system the mixture model includes the continuity equation and the momentum equations for the mixture, and the continuity equations for each dispersed phase. The slip velocities between the continuous phase and the dispersed phases are inferred from approximate algebraic balance equations. This reduces the computational effort considerably, especially when several dispersed phases are considered. Another common approach is the so called ’Lagrangian method’ which is mainly restricted to particulate suspensions. Within that approach only the fluid phase is treated as continuous while the motion of the discontinuous particulate phase is obtained by integrating the equation of motion of individual particles along their trajectories. (In practical applications a ”particle” may represent a single physical particle or a group of particles.) In this chapter we review the theoretical basis of multiphase flow dynamics. We first derive the basic equations of multiphase flows within the Eulerian approach using both volume averaging and ensemble averaging, and discuss the guiding principles for solving the closure problem within the Eulerian scheme. We then consider the basic formalism and applicability of the mixture model and the Lagrangian multiphase model. Finally, we give a few practical examples of possible closure relations.

11 1. Equations of multiphase ﬂow

1.2 Volume averaging

1.2.1 Equations

In this section we shall derive the ’equations of multiphase flow’ using the volume averaging method. To this end, we first define appropriate volume averaged dynamic flow quantities and then derive the required flow equations for those variables by averaging the corresponding phasial equations [Ish75, Dre83, Soo90, Dre71, DS71, Nig79]. While ensemble averaging may appear as the most elegant approach from the theoretical point of view, volume averaging provides perhaps the most intuitive and straightforward interpretation of the dynamic quantities and interaction terms involved. Vol- ume averaging also illustrates the potential problems and intricacies that are common to all averaging methods within Eulerian approach.

Volume averaging is based on the assumption that a length scale Lc exists such that l L L, where L is the ’macroscopic’ length scale of c the system and l is a length scale that we shall call ’mesoscopic’ in what follows. The mesoscopic length scale is associated with the distribution of the various phases within the mixture. (The ’microscopic’ length scale would then be the molecular scale.)

γ V = Vα + Vβ + Vγ Phase α Vα uα ρα

Aα τα pα

Phase β nˆαdAα Vβ Phase γ Vγ

Figure 1.1: Averaging volume V including three phases α, β, and γ.

To begin with, we consider a representative averaging volume V L3 ∼ c which contains distinct domains of each phase such that V = α Vα where V is the volume occupied by phase α within V (see Fig. 1.1). We as- α P sume that for each phase α the usual ﬂuid mechanical equations for mass, momentum and energy conservation are valid at any interior point of Vα,

12 1. Equations of multiphase ﬂow namely ∂ ρ + (ρ u ) = 0 (1.1) ∂t α ∇ · α α ∂ (ρ u ) + (ρ u u ) = p + τ + F (1.2) ∂t α α ∇ · α α α −∇ α ∇ · α α ∂ (ρ E ) + (ρ u E ) = (1.3) ∂t α α ∇ · α α α (u p ) + (u τ ) + u F J + J . −∇ · α α ∇ · α · α α · α − ∇ · qα Eα Here,

ρα = density of pure phase α

uα = ﬂow velocity

pα = pressure

τα = deviatoric stress tensor

Eα = total energy per unit mass

Fα = external force density

Jqα = heat ﬂux into phase α

JEα = heat source density. Eqns. (1.1) through (1.3) are assumed to be valid both for laminar and for turbulent flow. These equations are valid even if one of the phases is not actually a fluid but consists, e.g., of solid particles suspended in a fluid. In that case the stress tensor τα contains viscous stresses for the fluid and elastic deviatoric stresses for the particles. However, the concept of ’pressure’ may not always be very useful for a solid material. In such cases it may be preferable to use the total stress tensor σ = p 11 + τ instead, whence α − α α p + τ = σ . −∇ α ∇ · α ∇ · α Similarly to single phase flows, the energy equation (1.3) is necessary only in the presence of heat transfer. For simplicity, we shall from now on neglect the energy equation and consider only mass and momentum equations. For derivation of the energy equation for multiphase flows, see Refs. [Soo90] and [Hwa89]. Eqns. (1.1) and (1.2) for phase α are subject to the following boundary conditions at the interface Aαγ between phase α and any other phase γ inside volume V (see Fig. 1.2) [Soo90]. ρ (u u ) ˆn + ρ (u u ) ˆn = 0 (1.4) α α − A · α γ γ − A · γ ρ u (u u ) ˆn + ρ u (u u ) ˆn = (1.5) α α α − A · α γ γ γ − A · γ 2σαγ ( p 11 + τ ) ˆn + ( p 11 + τ ) ˆn σ + Rˆ , − α α · α − γ γ · γ − ∇A αγ R A | A| where

ˆnα = unit outward normal vector of phase α

13 1. Equations of multiphase ﬂow

α γ

nˆγ

dA nˆα

Figure 1.2: A portion of the interface between phases α and γ.

uA = velocity of the interface Rˆ = R / R A A | A| RA = inerface curvature radius vector

σαγ = interface surface tension = Rˆ = surface gradient operator ∇A ∇ − A · ∇ 11 = second rank unit tensor

The interface Aα = γ Aαγ may, however, have a very complicated shape which depends on time and which actually should be solved simultaneously S with the flow equations. Therefore, it is usually not possible to apply the boundary conditions (1.4) and (1.5) and to solve the mesoscopic equations (1.1) and (1.2) in the usual manner. This is the basic reason why we have to resort to averaged equations, in general. For any quantity qα (scalar, vector or tensor) defined in phase α we define the following averages [Ish75, Hwa89] 1 q = q dV (1.6) h αi V α ZVα 1 1 q˜ = q dV = q (1.7) α V α φ h αi α ZVα α ραqα dV ρ q q¯ = Vα = h α αi, (1.8) α ρ dV φ ρ˜ R Vα α α α where R φα = Vα/V. (1.9) is the volume fraction of phase α and is subject to the constraint that

φα = 1. (1.10) α X 14 1. Equations of multiphase ﬂow

The quantities defined by Eqns. (1.6), (1.7) and (1.8) are called the partial average, the intrinsic or phasic average and the Favréor mass weighted average of qα, respectively. At this point we leave until later the decision of which particular average of each flow quantity we should choose to appear as the final dynamic quantity of the averaged theory. In order to derive the governing equations for the averaged quantities defined above, we wish to apply averaging to equations (1.1) and (1.2). To this end, we notice that the following rules apply to the partial averages (and to the other two averages),

f + g = f + g (1.11) h i h i h i f g = f g (1.12) hh i i h ih i C = C for constant C. (1.13) h i It is also rather straightforward to show that the following rules hold for partial averages of various derivatives of qα [Soo90],

1 q = q + q nˆ dA (1.14) h∇ αi ∇h αi V α α ZAα 1 q = q + q nˆ dA (1.15) h∇ · αi ∇ · h αi V α · α ZAα ∂ ∂ 1 q = q q u nˆ dA. (1.16) h∂t αi ∂th αi − V α A · α ZAα

For later purposes, it is also useful to deﬁne the phase indicator function Θα such that 1, r phase α at time t Θ (r, t) = ∈ (1.17) α 0, otherwise. Using Eqns. (1.6) and (1.14) with qα = Θα, it is straightforward to see that

Θ = φ , (1.18) h αi α and that 1 nˆ dA = φ . (1.19) V α −∇ α ZAα Applying partial averaging on both sides of Eqns. (1.1) and (1.2) and using Eqns. (1.11)-(1.16) the following equations are obtained

∂ ρ + ρ u = Γ (1.20) ∂th αi ∇ · h α αi α ∂ ρ u + ρ u u = p + τ + F ∂th α αi ∇ · h α α αi −∇h αi ∇ · h αi h αi + Mα, (1.21)

15 1. Equations of multiphase ﬂow

where the so called ’transfer integrals’ Γα and Mα are deﬁned by

1 Γ = ρ (u u ) nˆ dA (1.22) α −V α α − A · α ZAα 1 M = ( p 11 + τ ) ˆn dA α V − α α · α ZAα 1 ρ u (u u ) nˆ dA. (1.23) −V α α α − A · α ZAα The flow equations as given by Eqns. (1.20) and (1.21) are not yet in a closed form amenable for solution. Firstly, the properties of each pure phase are not specified at this point. Secondly, the transfer integrals (1.22) and (1.23), which include the interactions (mass and momentum transfer) between phases, are still given in terms of integrals of the original mesoscopic quantities over the unknown phase boundaries. The additional constitutive relations, which are required to specify the material properties and to relate the transfer integrals with the proper averaged quantities, are discussed in more detail below. Thirdly, averages of various products of original variables that appear on the left side of the equations are independent of each other. Even if all the necessary constitutive relations are assumed to be known, we still have more independent variables than equations for each phase. In order to reduce the number of independent variables, we must express averages of these products in terms of products of suitable averages. This can be done in several alternative ways which may lead to slightly different results. Here we shall use Favréaveraging for velocity and, depending on which is more convenient, either partial or intrinsic averaging for density and pressure. Defining the velocity fluctuation δuα by

uα = u¯α + δuα, (1.24) it is easy to see that the averages of products that appear in Eqns. (1.20) and (1.21) can be written as

ρ u = ρ u¯ = φ ρ˜ u¯ (1.25) h α αi h αi α α α α ρ u u = ρ u¯ u¯ + ρ δu δu (1.26) h α α αi h αi α α h α α αi = φ ρ˜ u¯ u¯ + ρ δu δu α α α α h α α αi The averaged equations now acquire the form

∂ (φ ρ˜ ) + (φ ρ˜ u¯ ) = Γ (1.27) ∂t α α ∇ · α α α α ∂ (φ ρ˜ u¯ ) + (φ ρ˜ u¯ u¯ ) = ∂t α α α ∇ · α α α α (φ p˜ ) + τ + φ F˜ + M + τ , (1.28) −∇ α α ∇ · h αi α α α ∇ · h δαi 16 1. Equations of multiphase flow where τ = ρ δu δu . (1.29) h δαi −h α α αi This tensor is sometimes called a pseudo-turbulent stress tensor since it is analogous to the usual Reynolds stress tensor of turbulent one phase flow. Notice however, that tensor τ is defined here as a volume average instead h δαi of a time average as the usual Reynolds stress. It also contains momentum fluxes that arise both from the turbulent fluctuations of the mesoscopic flow and from the fluctuations of the velocity of phase α due to the presence of other phases. Consequently, tensor τ does not necessarily vanish even if h δαi the mesoscopic flow is laminar. Integrating the mesoscopic boundary conditions (1.4) and (1.5) over the interphase Aαγ , summing over α and γ and using definitions (1.22) and (1.23), we find that

Γα = 0 (1.30) α X 1 2σ M = ( σ + αγ Rˆ )dA. (1.31) α −2V −∇A αγ R A α α,γ Aαγ A X αX=γ Z | | 6 Eqn. (1.30) ensures conservation of the total mass of the mixture, while the right side of Eqn. (1.31) gives rise to surface effects such as ’capillary’ pressure differences between various phases. Eqns. (1.27) and (1.28) together with constraints (1.10), (1.30) and (1.31) are the most general averaged equations of multiphase flow (with no heat transfer), which can be derived without reference to the particular properties of the system (other than the general continuum assumptions). The basic dynamical variables of the averaged theory can be taken to be the three components of the mass-averaged velocities u¯α and the volume fractions φ (or, alternatively, the averaged densities ρ ). Provided that α h αi all the other variables and terms that appear in Eqns. (1.27) and (1.28) can be related to these basic variables using definitions (1.6) through (1.8), constraints (1.10), (1.30) and (1.31) and constitutive relations, we thus have a closed set of four unknown variables and four independent equations for each phase α.

1.2.2 Constitutive relations Eqns. (1.27) and (1.28) are, in principle, exact equations for the averaged quantities. So far, they do not contain much information about the dynamics of the particular system to be described. That information must be provided by a set of system dependent constitutive relations which specify the material properties of each phase, the interactions between diﬀerent phases and the (pseudo)turbulent stresses of each phase in the presence of other phases.

17 1. Equations of multiphase ﬂow

These relations ﬁnally render the set of equations in a closed form where solution is feasible. At this point we do not attempt to elaborate in detail the possible strategies for attaining the constitutive relations in speciﬁc cases, but simply state the basic principles that should be followed in inferring such relations. The unknown terms that appear in the averaged equations (1.27) and (1.28), such as the transfer integrals and stress terms that still contain mesoscopic quantities, should be replaced by new terms which depend only on the averaged dynamic quantities (and their deriva- • tives),

have the same physical content, tensorial form and dimension as the • original terms,

have the same symmetry properties as the original terms (isotropy, • frame indiﬀerence etc.),

include the effects of all the physical processes or mechanisms that are • considered to be important in the system to be described. Typically, constitutive relations are given in a form where these new terms include free parameters which are supposed to be determined experimentally. For more detailed discussion of the constitutive relations and constitutive principles, see Refs. [DALJ90, Dre83, DLJ79, Dre76, Hwa89, HS89, HS91, BS78, Buy92a, Buy92b]. In some cases constitutive laws can readily be derived from the properties of the mixture, or from the properties of the pure phase. For example, the incompressibility of the pure phase α implies the constitutive relation ρ˜α=constant. Similarly, the equation of state pα = Cρα, where C=constant for the pure phase, impliesp ˜α = Cρ˜α. In most cases, however, the constitutive relations must be either extracted from experiments, derived analytically under suitable simplifying assumptions, or postulated. Including a given physical mechanism in the model by imposing proper constitutive relations is not always straightforward even if adequate experimental and theoretical information is available. In particular, making specific assumptions concerning one of the unknown quantities may induce constraints on other terms. For example, the transfer integrals Γα and Mα contain the effect of exchange of mass and momentum between the phases. According to Eqn. (1.22), the quantity Γα gives the rate of mass transfer per unit volume through the phase boundary Aα into phase α from the other phases. In a reactive mixture, where phase α is changed into phase γ, the mass transfer term Γα might be given in terms of the experimental rate of the chemical reaction α γ, correlated to the volume fractions φ and φ , → α γ and to the temperature of the mixture T . Similarly, the quantity Mα gives the rate of momentum transfer per unit volume into phase α through the

18 1. Equations of multiphase ﬂow

phase boundary Aα. The second integral on the right side of Eqn. (1.23) contains the transfer of momentum carried by the mass exchanged between phases. It is obvious that this part of the momentum transfer integral Mα must be consistently correlated with the mass transfer integral Γα. Simi- larly, the first integral on the right side of Eqn. (1.23) contains the change of momentum of phase α due to stresses imposed on the phase boundary by the other phases. Physically, this term contains forces such as buoyancy which may be correlated to average pressures and gradients of volume fractions, and viscous drag which might be correlated to volume fractions and average velocity differences. For instance in a liquid-particle suspension, the average stress inside solid particles depends on the hydrodynamic forces acting on the surface of the particles. The choice of, e.g., drag force correlation between fluid and particles should therefore influence the choice of the stress correlation for the particulate phase. While this particular problem can be solved exactly for some idealized cases [DALJ90], there seems to be no general solution available. Perhaps the most intricate term which is to be correlated to the averaged quantities through constitutive relations is the tensor τ given by h δαi Eqn. (1.29). It contains the momentum transfer inside phase α which arises from the genuine turbulence of phase α and from the velocity fluctuations due to presence of other phases, and which are present also in the case that the flow is laminar in the mesoscopic scale. Moreover, the truly turbulent fluctuations of phase α may be substantially modulated by the other phases. Bearing in mind the intricacies that are encountered in modeling turbulence in single phase flows, it is evident that inferring realistic constitutive relations for tensor τ remains as a considerable challenge. It may, h δαi however, be attempted, e.g., for fluid-particle suspensions by generalising the corresponding models for single phase flows, such as turbulence energy dissipation models, large-eddy simulations or direct numerical simulations. A recent review on the topic is given by Crowe, Troutt and Chung in Ref. [CTC96]. In the remaining part of this section we shall shortly discuss a few particular cases where additional simplifying assumptions can be made, namely liquid-particle suspension, bubbly flow and multifluid flow. These examples emphasize further the circumstance that no general set of equations exists that, as such, would be valid and readily solvable for an arbitrary multiphase flow, or even for an arbitrary two-phase flow. Instead, the flow equations appropriate for each particular system should be derived separately starting from the general (but unclosed) set of equations given in section 1.2 and utilizing all the specific assumptions and approximations that are plausible for that system (or class of systems). We notice, however, that the assumptions made here concerning, e.g., bubbly flow may not be generally valid for all such flows. Nor are they the only possible extra assumptions that can be made, but should be taken merely as examples of the kind of hypotheses

19 1. Equations of multiphase ﬂow that are reasonable owing to the nature of that category of systems. De- tailed examples of more complete closure relations will be given in section 1.6. In chapter 2 (see section 2.5) we shall brieﬂy discuss novel numerical methods that can be used to infer constitutive relations by means of direct numerical simulation in a mesoscopic level.

Liquid-particle suspension Consider a binary system of solid particles suspended in a Newtonian liquid. We denote the continuous ﬂuid phase by subscript f and the dispersed particle phase by subscript d. We assume that both phases are incompressible, that the suspension is non-reactive, i.e., there is no mass transfer between the two phases, and that surface tension between solid and liquid is negligible. Both the densitiesρ ˜f andρ ˜d are thus constants, and

Γf = Γd = 0 (1.32)

Mf + Md = 0. (1.33) The mutual momentum transfer integral can now be written as 1 M M = M = ( p 11 + τ ) ˆn dA ≡ f − d V − f f · f ZAf 1 = ( p 11 + τ ) ˆn dA, (1.34) −V − f f · ZA where A = A = A and ˆn = ˆn = ˆn . Introducing the fluid pressure f d d − f fluctuation by δp = p p˜ and using Eqn. (1.19), the momentum transfer f f − f integral can be cast in the form M =p ˜ φ + D, (1.35) f ∇ where 1 D = ( δp 11 + τ ) ˆn dA, (1.36) −V − f f · ZA and φ = φf . The averaged flow equations can now be written in the final form as ∂ φ + (φu¯ ) = 0 (1.37) ∂t ∇ · f ∂ (1 φ) + ((1 φ)u¯ ) = 0 (1.38) ∂t − ∇ · − d ∂ ρ˜ [ (φu¯ ) + (φu¯ u¯ )] = φ p˜ + τ + φF˜ f ∂t f ∇ · f f − ∇ f ∇ · h f i f +D + τ (1.39) ∇ · h δf i ∂ ρ˜ [ ((1 φ)u¯ ) + ((1 φ)u¯ u¯ )] = + σ + (1 φ)F˜ (1.40) d ∂t − d ∇ · − d d ∇ · h di − d D p˜ φ + τ , − − f ∇ ∇ · h δdi 20 1. Equations of multiphase flow where τ is the averaged viscous stress tensor of the fluid, and σ is h f i h di the averaged total stress tensor of the dispersed phase. An example of more complete constitutive relations for a dilute liquid-particle suspension is given in section 1.6.1.

Bubbly ﬂow with mass transfer If the dispersed phase consists of small gas bubbles instead of solid particles, the overall structure of the system still remains similar to the liquid-particle suspension discussed in the previous section. A few things will change, however. Firstly, the dispersed phase is not incompressible. Instead, the intrinsic densityρ ˜d depends on pressurep ˜d and temperature T as given by the equation of state of the gas,

ρ˜d =ρ ˜d(˜pd,T ). (1.41)

Secondly, mass transfer between the phases generally occur. The gas phase usually consists of several gaseous components including the vapor of the liquid. The mass transfer may take place as evaporation of the liquid or dissolution of the gas at the surface of the bubbles. Instead of Eqn. (1.32) we now have Γ = Γ = Γ, (1.42) f − d where the mass transfer rate Γ is a measurable quantity which may depend on the pressure, on the temperature and on the prevailing vapor content of the gas etc. In this case, also the second term on the right side of Eqn. (1.23), which includes the momentum carried by the mass exchanged, is non-zero. This term can be related to the mass transfer rate Γ by deﬁning the average velocity u¯m at the phase interface by the equation 1 ρ u (u u ) nˆ dA = u¯ Γ. (1.43) V d d d − A · d m ZAd

At this stage, velocity u¯m is of course unknown and must be modeled separately. A natural ﬁrst choice would be that u¯m is the mass averaged velocity of the mixture, i.e.,

φρ˜ u¯ + (1 φ)˜ρ u¯ u¯ = f f − d d . (1.44) m φ ρ˜ + (1 φ)˜ρ f − d Thirdly, surface tension between the phases may be important. For small bubbles one may ignore the surface gradient term σ in Eqn. (1.31). −∇A df Assuming that 2σ / R =p ˜ p˜ (capillary pressure) it is straightforward df | A| d − f to see that, instead of Eqn. (1.33), we now have

M + M = (˜p p˜ ) φ. (1.45) f d − d − f ∇ 21 1. Equations of multiphase ﬂow

Following the analysis given by Eqns. (1.34) through (1.35) we can now verify that M =p ˜ φ + D + u¯ Γ (1.46) f f ∇ m d M =p ˜ (1 φ) D u¯ Γ , (1.47) d d ∇ − − − m d where the quantity D is still given by Eqn. (1.36). The ﬂow equations for bubbly ﬂow with small bubbles may thus be given in the form ∂ ρ˜ φ + ρ˜ (φu¯ ) = Γ (1.48) f ∂t f ∇ · f ∂ [(1 φ)˜ρ ] + [(1 φ)˜ρ u¯ ] = Γ (1.49) ∂t − d ∇ · − d d −

∂ ρ˜ [ (φu¯ ) + (φu¯ u¯ )] = φ p˜ + τ (1.50) f ∂t f ∇ · f f − ∇ f ∇ · h f i +φF˜ + D + u¯ Γ + τ f m d ∇ · h δf i ∂ [(1 φ)˜ρ u¯ ] + [(1 φ)˜ρ u¯ u¯ ] = (1 φ) p˜ + τ (1.51) ∂t − d d ∇ · − d d d − − ∇ d ∇ · h di +(1 φ)F˜ D u¯ Γ + τ . − d − − m d ∇ · h δdi Multifluid system As a final example, we consider a system which consists of several continuous or discontinuous fluid phases under the simplifying assumptions that there is no mass transfer between the phases and that the surface tension can be neglected for each pair of phases. It thus follows that

Γα = 0 for all phases α (1.52)

Mα = 0. (1.53) α X Analogously to Eqn. (1.35) we can decompose the momentum transfer integrals as M =p ˜ φ + D , (1.54) α α ∇ α α where 1 D = ( δp 11 + τ ) ˆn dA. (1.55) α V − α α · α ZAα In the special case where the system is at complete rest, the phases must share the same pressure (this follows from Eqn. (1.5) in the case that σαγ = 0). We shall assume here that this is approximately true also in the general case where ﬂow is present. We thus have thatp ˜ p˜, wherep ˜ is the common α ≈ pressure of all phases. Since ( φ ) = 0, it follows from Eqns. (1.53) ∇ α α and (1.54) that P Dα = 0. (1.56) α X 22 1. Equations of multiphase ﬂow

For the present multiﬂuid system, Eqns. (1.27) and (1.28) can now be written in the form

∂ (φ ρ˜ ) + (φ ρ˜ u¯ ) = 0 (1.57) ∂t α α ∇ · α α α ∂ (φ ρ˜ u¯ ) + (φ ρ˜ u¯ u¯ ) = ∂t α α α ∇ · α α α α φ p˜ + τ + φ F˜ + D + τ . (1.58) − α∇ ∇ · h αi α α α ∇ · h δαi These equations will be further considered in chapter 2 where various methods for numerical solution of multiphase flows are discussed. The apparent restriction to a single common pressurep ˜ of phases in Eqns. (1.58) is not actually a severe limitation of generality. If for any reason the pressure varies between phases, it is always possible to define p˜ as an appropriate average of the phasial pressuresp ˜α and to include the effect of the deviatoric partp ˜ p˜ in the other terms on the right side of α − Eqn. (1.58) such as in τ or in D . ∇ · h αi α

1.3 Ensemble averaging

In the previous section we derived the equations of multiphase flow (ignoring the energy equation) using volume averaging. In this section we shall repeat the derivation using a different approach, namely ensemble averaging [Ish75, Dre83, Buy71, Hwa89, JL90]. As we shall see, the resulting general equations are formally identical to those derived in the previous section, Eqns. (1.27) and (1.28). While ensemble averaging, of all the averaging methods that are commonly used within the Eulerian approach, appears as the most elegant one, mathematical charm alone would not be a sufficient cause for duplicating our efforts at this point. As discussed in section 1.2.2 however, the major problem within multiphase fluid dynamics is not derivation of the conservation equations, but the closure of the equations. Ensemble averaging, as the standard averaging method of also the modern statistical physics, provides a different view in the physical interpretation of the interaction terms and of the Reynolds stresses and may thereby provide a different approach for solving the closure problem. Conceptually, ensemble averaging is achieved by repeating the measurement at a fixed time and position for a large number of systems with identical macroscopic properties and boundary conditions, and finding the mean value of the results. Although the properties and the boundary conditions are unchanged at the macroscopic level for each system, they differ at the mesoscopic level. This leads to a scatter of the observed values. We denote the collection of these macroscopically identical systems by C (the ensemble) and its individual member by µ. If f(r, t; µ) is any quantity observed

23 1. Equations of multiphase flow for a system µ at point r and time t, its ensemble average f is defined by h i f (r, t) = f(r, t; µ) dm(µ), (1.59) h i ZC where the measure dm(µ) is the probability of observing system µ within C. If all the necessary derivatives of f exist, it follows from the linearity of the ensemble averaging that ∂ ∂ f = f (1.60) h∂t i ∂th i f = f . (1.61) h∇ i ∇h i Notice that Eqns. (1.60) and (1.61) do not include any additional interfacial terms, in contrast to Eqns. (1.14) through (1.16) for volume averaging. The difference lies in the different definitions of variables and averages. The observable quantity f itself is not associated to any particular phase of the system. For example if f is density, then f(r, t; µ) is the local value of the density of the phase that happens to occupy point r at time t. Also the ensemble average is calculated without paying any attention to the phase that occupies the point at which the average is calculated. The volume average of a quantity defined for a certain phase is, on the other hand, calculated only over the part of the total averaging volume occupied by that particular phase, which gives rise to the surface integrals. The partial average of f in phase α is defined by

Θ f , (1.62) h α i where Θα is the phase indicator function deﬁned by Eqn. (1.17). Consider- ing the phase indicator function as a generalized function (distribution), it can be shown that it satisﬁes the equation DsΘ ∂ α Θ + u Θ = 0, (1.63) Dt ≡ ∂t α A · ∇ α To see this, consider ∂ Θα + uA Θα ψ dV dt (1.64) R3 R ∂t · ∇ Z × ∂ = Θα ψ + (ψuA) dV dt − R3 R ∂t ∇ · Z × ∞ ∂ = ψ + (ψuA) dV dt − Vα ∂t ∇ · Z−∞ Z d = ∞ ψ dV dt − dt Vα(t) Z−∞ Z ! = 0,

24 1. Equations of multiphase ﬂow

where uA is the velocity of the phase interface, Vα(t) is the volume occupied by phase α at time t, and ψ is a test function, which is suﬃciently smooth and has a compact support both in V and t. In order that the second line makes sense we must extend uA smoothly through phase α. It is now easy to show that

Θ f = Θ f f Θ (1.65) h α∇ i ∇h α i − h ∇ αi Θ f = Θ f f Θ (1.66) h α∇ · i ∇ · h α i − h · ∇ αi ∂ ∂ ∂ Θ f = Θ f f Θ (1.67) h α ∂t i ∂th α i − h ∂t αi ∂ = Θ f + fu Θ , ∂th α i h A · ∇ αi where on the last line we have utilized Eqn. (1.63). The gradient of the phase indicator function is non-zero only at the phase interfaces, which leads to the conclusion that Eqns. (1.65) through (1.67) are counterparts of Eqns. (1.14) through (1.16) and, in particular, that the second terms on the right side of Eqns. (1.65) through (1.67) are counterparts of the surface integrals in Eqns. (1.14) through (1.16). We assume that inside each phase, the normal fluid mechanical equations for mass and momentum are valid, namely ∂ ρ + (ρu) = 0 (1.68) ∂t ∇ · ∂ (ρu) + (ρuu) = σ + F. (1.69) ∂t ∇ · ∇ · where ρ, u, τ and F are local density, flow velocity, stress tensor and external force density, respectively. In what follows, we do not consider the energy equation. The hydrodynamic quantities that appear in Eqns. (1.68) and (1.69) are assumed to be well behaving within each phase but can have discontinuities at phase interfaces. Multiplying Eqns. (1.68) and (1.69) by Θα, performing ensemble averaging and using Eqns. (1.65) and (1.67), we get ∂ Θ ρ + Θ ρu = ρ(u u ) Θ (1.70) ∂th α i ∇ · h α i h − A · ∇ αi ∂ Θ ρu + Θ ρuu = Θ σ + Θ F (1.71) ∂th α i ∇ · h α i ∇ · h α i h α i + (ρu(u u ) σ) Θ . h − A − · ∇ αi In analogy with Eqns. (1.7) and (1.8) we define the intrinsic and Favré averages of any quantity f as

f˜ = Θ f / Θ = Θ f /φ (1.72) α h α i h αi h α i α f¯ = Θ ρf / Θ ρ = Θ ρf /(φ ρ˜ ), (1.73) α h α i h α i h α i α α 25 1. Equations of multiphase flow respectively. Here, φ = Θ , (1.74) α h αi which we call the ’volume fraction’ of phase α following the common convention even though ’statistical fraction’ might be a more proper term. In order to approach a closed set of equations we again use Favréaver- aged velocity u¯α and the velocity fluctuation δuα defined for phase α as

u¯ = Θ ρu /(φ ρ˜ ) (1.75) α h α i α α δu = u u¯ . (1.76) α − α With these conventions the momentum ﬂux term in Eqn. (1.71) can be rewritten as Θ ρuu = φ ρ˜ u¯ u¯ τ , (1.77) h α i α α α α − δα where the Reynolds stress tensor τδα is deﬁned by

τ = Θ ρδu δu . (1.78) δα −h α α αi Using Eqns. (1.74) through (1.77), the averaged equations (1.70) and (1.71) ﬁnally acquire the form

∂ (φ ρ˜ ) + (φ ρ˜ u¯ ) = Γ (1.79) ∂t α α ∇ · α α α α ∂ (φ ρ˜ u ) + (φ ρ˜ u¯ u¯ ) ∂t α α α ∇ · α α α α = (φ σ˜ + τ ) + (φ F˜ ) + M , (1.80) ∇ · α α δα α α α where the quantities Γα and Mα are deﬁned by

Γ = ρ(u u ) Θ (1.81) α h − A · ∇ αi M = (ρu(u u ) σ) Θ . (1.82) α h − A − · ∇ αi These terms are the analogues of the transfer integrals that appear in the corresponding volume averaged equations (1.22) and (1.23). They include the rate of mass and momentum transfer between the phases. As stated before, the general averaged equations obtained using ensemble averaging are formally identical with those derived within the volume averaging scheme in section 1.2. All the terms that appear in the ensemble averaged equations have analogous physical content with the corresponding term in the volume averaged equations. The only difference is that the different formal definitions of terms such as τδα,Γα and Mα within these two approaches may offer different ways of relating these quantities with the basic averaged variables φα,ρ ˜α,p ˜α, u¯α, etc., and thereby solving the closure problem for a given system.

26 1. Equations of multiphase ﬂow

1.4 Mixture models

The mixture model (or algebraic slip model) is a simplified formulation of the multiphase flow equations. We consider a suspension of a dispersed phase (particles, drops, or bubbles) in a continuous fluid (liquid or gas). If the dispersed phase follows closely the fluid motion (small particles), it seems natural to write the balance equations for the mixture of the dispersed and continuous phases and take the relative motion of the phases into account as a correction. The mixture model consists then of the continuity and momentum equations for the mixture and the continuity equations for the individual dispersed phases. The slip velocity between the dispersed and continuous phases is taken into account by introducing corresponding convection terms in the continuity equations. The essential character of the mixture model is that only one set of velocity components is solved from the differential equations for momentum conservation. The velocities of the dispersed phases are inferred from approximate algebraic balance equations. This reduces the computational effort considerably, especially when several dispersed phases need to be considered. The mixture model equations are derived in the literature applying various approaches [Ish75, Ung93, Gid94]. The form of the equations also varies depending on the application. Ishii [Ish75] derives the mixture equations from a general balance equation. In this section, we derive the mixture model equations from the original multiphase equations. This approach is transparent and the required simplifications are clearly shown. Furthermore, the applicability of the model can be explicitly analysed.

Continuity equation for the mixture From the continuity equation for phase α (1.27), we obtain by summing over all phases n n n ∂ (φ ρ˜ ) + (φ ρ˜ u¯ ) = Γ (1.83) ∂t α α ∇ · α α α α α=1 α=1 α=1 X X X The right hand side of Eqn. (1.83) vanishes due to the conservation of the total mass, Eqn. (1.30), and we obtain the continuity equation of the mixture ∂ (ρ ) + (ρ u¯ ) = 0 (1.84) ∂t m ∇ · m m Here the mixture density and the mixture velocity are deﬁned as n ρm = φαρ˜α (1.85) α=1 X n n 1 u¯ = φ ρ˜ u¯ = c u¯ (1.86) m ρ α α α α α m α=1 α=1 X X 27 1. Equations of multiphase ﬂow

The mixture velocity u¯m represents the velocity of the mass center. Notice that ρm varies although the component densities are constant. The mass fraction of phase α is deﬁned as

φαρ˜α cα = (1.87) ρm Eqn. (1.84) has the same form as the continuity equation for single phase ﬂow.

Momentum equation for the mixture The momentum equation for the mixture follows from the phase momentum equations (1.28) by summing over all phases

n n ∂ φ ρ˜ u¯ + φ ρ˜ u¯ u¯ ∂t α α α ∇ · α α α α α=1 α=1 X nX n = (φ p˜ ) + φ (˜τ +τ ˜ ) − ∇ α α ∇ · α α δα α=1 α=1 Xn n X + φαF˜ α + Mα. (1.88) α=1 α=1 X X Here, the stress terms have been written in terms of intrinsic averages of the stress tensors using Eqn. (1.7). Using the deﬁnitions (1.85) and (1.86) of the mixture density ρm and the mixture velocity u¯m, the second term of (1.88) can be rewritten as

n n φ ρ˜ u¯ u¯ = (ρ u¯ u¯ ) + φ ρ˜ u¯ u¯ (1.89) ∇ · α α α α ∇ · m m m ∇ · α α mα mα αX=1 αX=1 where u¯mα is the diﬀusion velocity, i.e., the velocity of phase α relative to the center of the mixture mass

u¯ = u¯ u¯ (1.90) mα α − m In terms of the mixture variables, the momentum equation takes the form ∂ (ρ u¯ ) + (ρ u¯ u¯ ) = p + ( τ + τ ) + τ ∂t m m ∇ · m m m −∇ m ∇ · h mi h δmi ∇ · h Dmi +Fm + Mm (1.91)

The three stress tensors are deﬁned as n τ = φ τ˜ (1.92) h mi − α α α=1 X 28 1. Equations of multiphase ﬂow

n τ = φ ρ˜ δu δu (1.93) h δmi − αh α α αi αX=1 n τ = φ ρ˜ u¯ u¯ (1.94) h Dmi − α α mα mα αX=1 and represent the average viscous stress, turbulent stress, and diﬀusion stress due to the phase slip, respectively. In Eqn. (1.91), the pressure of the mixture is deﬁned by the relation

n pm = φαp˜α (1.95) αX=1 In practice, the phase pressures are often taken to be equal, i.e.,p ˜α = pm. Accordingly, the last term on the right hand side of (1.91), Mm, comprises only the inﬂuence of the surface tension force on the mixture and depends on the geometry of the interface. The other additional term in (1.91) compared to the one phase momentum equation is the diﬀusion stress term τ ∇ · Dm representing the momentum transfer due to the relative motions.

Continuity equation for a phase We return to consider an individual phase. Using the deﬁnition of the diﬀu- sion velocity (1.90) to eliminate the phase velocity in the continuity equation (1.27) gives

∂ (φ ρ˜ ) + (φ ρ˜ u¯ ) = Γ (φ ρ˜ u¯ ) (1.96) ∂t α α ∇ · α α m α − ∇ · α α mα If the phase densities are constants and phase changes do not occur, the continuity equation reduces to ∂ φ + (φ u¯ ) = (φ u¯ ) (1.97) ∂t α ∇ · α m −∇ · α mα The term on the right hand side represents the diﬀusion of the particles due to the phase slip.

Diﬀusion velocity In the mixture model, the momentum equations for the dispersed phases are not solved and therefore a closure model has to be derived for the diﬀusion velocities. The balance equation for calculating the relative velocity can be rigorously derived by combining the momentum equations for the dispersed phase and the mixture. In the following, we consider one dispersed particulate phase, p, for simplicity. Using Eqn. (1.54) for Mp and the continuity equation, the momentum equation of the dispersed phase p (1.28) can be

29 1. Equations of multiphase ﬂow

rewritten as follows (here gravity is used for external force , i.e., F˜ α =ρ ˜αg, Fm = ρmg) ∂ φ ρ˜ u¯ + φ ρ˜ (u¯ )u¯ = φ (˜p ) + [φ (˜τ +τ ˜ )] p p ∂t p p p p · ∇ p − p∇ p ∇ · p p δp + φpρ˜pg + Dp (1.98)

The corresponding equation for the mixture is ∂ ρ u¯ + ρ (u¯ )u¯ = p + (τ + τ + τ ) + ρ g (1.99) m ∂t m m m · ∇ m −∇ m ∇ · m δm Dm m

Here we have neglected the surface tension forces and therefore Mm = 0. Assuming that the phase pressures are equal, i.e., pm =p ˜p, we can eliminate the pressure gradient from (1.98) and (1.99). As a result we obtain an equation for Dp

∂ ∂ D = φ ρ˜ u¯ + (˜ρ ρ ) u¯ p p p ∂t mp p − m ∂t m +φ [ρ˜ (u¯ )u¯ ρ (u¯ )u¯ ] p p p · ∇ p − m m · ∇ m [φ (˜τ +τ ˜ )] + φ (τ + τ + τ ) −∇ · p p δp p∇ · m δm Dm φ (˜ρ ρ ) g (1.100) − p p − m In (1.100) we have utilized the definition (1.90) for the diffusion velocity u¯mp. Next, we will make several approximations to simplify Eqn. (1.100). Using the local equilibrium approximation, we drop from the first term the time derivative of u¯mp. In the second term, we approximate

(u¯ )u¯ (u¯ )u¯ (1.101) p · ∇ p ≈ m · ∇ m The viscous and diffusion stresses are omitted as small compared to the leading terms. The turbulent stress cannot be neglected if we wish to keep the turbulent diffusion of the dispersed phase in the model. However, all turbulent effects are omitted for the moment. In that case, the final simplified equilibrium equation for Dp is

∂ Dp = φp(˜ρp ρm) g (u¯m )u¯m u¯m . (1.102) − " − · ∇ − ∂t #

Since D is a function of the slip velocity u¯ = u¯ u¯ , Eqn. (1.102) is an p cp p − c algebraic formula for the diﬀusion velocity

u¯ = (1 c )u¯ (1.103) mp − p cp The mixture model consists of Eqns. (1.84), (1.91), (1.96), and (1.102) together with constitutive equations for the viscous and turbulent stresses.

30 1. Equations of multiphase ﬂow

Validity of the mixture model The terms omitted in arriving to Eqn. (1.102) from Eqn. (1.100) (except turbulence terms) can be rewritten in the following form

∂ φpρ˜p u¯mp + (u¯mp )u¯mp + φpρ˜p[(u¯m )u¯mp + (u¯mp )u¯m] "∂t · ∇ # · ∇ · ∇ +φ (τ + τ ) (φ τ˜ ) (1.104) p∇ · m Dm − ∇ · p p The local equilibrium approximation requires that the particles are rapidly accelerated to the terminal velocity. This corresponds to setting the ﬁrst term in (1.104) equal to zero. Consider ﬁrst a constant body force, like gravitation. A criterion for neglecting the acceleration is related to the relaxation time of a particle, tp. In the Stokes regime tp is given by

2 ρ˜pdp tp = , Rep < 1 (1.105) 18µm and in the Newton regime (constant CD) by

4˜ρpdp tp = , Rep > 1000 (1.106) 3˜ρcCDut where ut is the terminal velocity. Within the time tp, the particle travels the distance lp = tput/e, which characterizes the length scale of the acceleration. If the density ratioρ ˜p/ρ˜c is small, the virtual mass and Basset terms in the equation of motion cannot be neglected. The Basset term in particular eﬀectively increases the relaxation time. The true length scale lp0 of the particle acceleration can be an order of magnitude larger than lp [MTK96]. An appropriate requirement for the local equilibrium is thus lp0 << L, where L is a typical dimension of the system. The second term in (1.104) corresponds, in rotational motion, the Cori- olis force. The radial particle velocity caused by the centrifugal acceleration causes in turn a tangential acceleration. To the ﬁrst order, the second term in (1.104) is proportional tou ¯mϕu¯cp/r, which has to be compared with the 2 leading termu ¯mϕ/r (r is the radius of curvature). Neglecting the second term thus requires simply that u¯ cp << 1 (1.107) u¯mϕ In the Stokes regime, this condition can be expressed as

18µ d << c , (1.108) p ω˜ (ρ ρ˜ ) s p − c where ω is the angular velocity of the rotation.

31 1. Equations of multiphase ﬂow

In the last two terms of Eqn. (1.104), the viscous stresses can obvi- ously be regarded as small compared to the leading terms, except possibly inside a boundary layer. The diffusion stress can be neglected within the approximation of local equilibrium. In the above analysis, we assumed that the suspension is homogeneous in small spatial scales. If this is not the case and dense clusters of particles are formed, the mixture model is usually not applicable. Clustering in a scale comparable to the length scale of turbulent fluctuations is typical for small particles (dp < 200µm) in gases. The clustering can lead to a substantial decrease in the effective drag coefficient. Consequently, the particle relaxation time becomes large and the local equilibrium approximation is not valid. Although the mixture model is in principle valid for small particles (dp < 50µm) in gases, it can be used only for dilute suspensions with solids to gas mass ratio below 1.

1.5 Particle tracking models

Historically Lagrangian tracking models were first introduced in very dilute flows of particulate suspensions[BH79], characteristic e.g. for electrostatic precipitators. In these flows, the dispersed phase number density is low enough such that the flow is dominated by the continuous carrier phase. Mathematically the delineation between dilute and dense particulate flows is established by the hydrodynamic response (or relaxation) time of the particle tp and the mean time between successive collisions between the particles tc. In a dilute flow tp/tc < 1. The particle then has enough time to respond to the surrounding fluid field before the next collision, and the motion of the particle is primarily controlled by the fluid flow. This assumption allows separation of the solution for the two phases. Within the most simple approach to particle tracking one first solves the flow of the carrier fluid without taking particles into consideration. Next, particles are released at desired positions in the solution domain. The trajectories of the particles are then found by integrating an appropriate force balance equation (see section 1.5.1) for each particle. Particle patches instead of single particles may also be used at this point in order to reduce computational effort. This type of approach is also known as one-way coupling as the information is mainly transfered from the carrier phase to the dispersed phase and not in the opposite direction. In flows frequently found, e.g., in pneumatic conveying, the number density of the particulate phase is relatively high such that the presence of particles affects the flow of the carrier phase while the collisions between particles can still be ignored. Then, two-way coupling is said to prevail since information is transfered from the carrier phase to dispersed phase and vice versa. Lagrangian particle tracking method can be used to solve also this

32 1. Equations of multiphase flow type of flow through an appropriate iterative procedure. The principle of numerical solution for one-way coupled systems and for two-way coupled systems is illustrated in Fig. 2.5. In dense particulate flows, such as those found in fluidized beds, we have that tp/tc > 1, whereby the motion of a particle is significantly affected also by interactions with other particles. These flows represent the case of four- way coupling where information is also transfered between particles, and are often solved using Eulerian multiphase models. As stated above, particle tracking method includes first solving the usual single-phase flow equations with proper boundary conditions for the carrier phase, and then an initial value problem for an ordinary differential equation separately for each particle. (For flows with two-way coupling, this procedure must be iterated.) In many cases particle tracking method leads to a marked simplification as compared to the continuum Eulerian method which requires the solution of a boundary value problem for coupled partial differential equations, (Eqns. (1.27) and (1.28)), for both phases. Funda- mental difficulties associated with the closure of Eulerian multiphase models may also be avoided within the Lagrangian approach. Although the particle tracking method is conceptually simple, it is not always quite straightforward in practice. Firstly, the equation of motion of an individual particle in the surrounding fluid, discussed in the next section, may be quite complicated. Secondly, turbulent flow poses a problem also within particle tracking method. In particular, the modification of fluid turbulence due to presence of the dispersed phase still lacks models with adequate theoretical justifica- tion [Cro93]. Provided that an adequate equation of motion of the particle is given, the particle tracking method is in general readily applicable in laminar flows and in cases where only the mean trajectories of particles in a turbulent flow are of interest. If, however, dispersion of particles in a turbulent flow is considered, additional modeling is needed in order to relate the dispersion rate of particles with the statistical characteristics of turbulence of the carrier fluid. That topic is discussed in section 1.5.2.

1.5.1 Equation of motion for a single particle

The basic equation that describes the motion of a sphere settling in a qui- escent ﬂuid due to gravity is the well-known equation by Basset [Bas88], Boussinesq [Bou03] and Oseen [Ose27] (BBO). The original BBO equation was based on the assumption that the Reynolds number of the particle is low enough for the disturbance ﬁeld produced by the motion of the sphere to be governed by the unsteady Stokes equation. Later, Tchen extended the

33 1. Equations of multiphase ﬂow

BBO equation to an unsteady and non-uniform ﬂow as follows [Tch47]

III I II dv m m d m = 6πr µ (u v) f p + f (u v) p dt p f − − ρ ∇ 2 dt − z f}| { z }| { z }| { IV t d 2 dτ u(y(τ), τ) v(τ) + 6r √πµfρf { − } dτ z p }| √t τ { Zt0 − V + (m m )g . (1.109) p − f Here, rp is the radius of thez particle,}| {mp and mf are the mass of the particle and the mass of a fluid sphere of radius rp, ρf and µf are the density and the dynamic viscosity of the fluid, v(t) is the velocity vector of the particle instantaneously centred at y(t) and u(y(t), t) is the Eulerian velocity vector of the fluid at position y(t). It should be noted that u represents the fluid velocity at the mass centre of the particle as if the particle would not create any disturbance field. The convective derivative following the motion of the particle is given by d ∂ ∂ = + v . (1.110) dt ∂t j ∂x j y(t) The numbered terms in Eqn. (1.109) are the Stokes drag force (I), the force by fluid pressure gradient (II), the force by added mass (III), the Basset history term (IV) and the buoyancy term (V). Notice, that hydrostatic pressure component is not included in the pressure p, but is taken into account in the buoyancy term V in Eqn. (1.109). Several authors have pointed out inconsistencies in Eqn. (1.109) [Lum57, Buy66, Ril71]. Maxey and Riley [MR83] were the first to derive the equation of motion of a small particle rationally from basic principles. They formulated the problem for the motion of a rigid Stokes sphere in a nonuniform flow field following the approach of Riley [Ril71] for the undisturbed flow. The disturbance field caused by the sphere was calculated generalizing the results of Basset [Bas88] and extending the work of Burgers [Bur38]. The equation of motion given by Maxey and Riley is dv du m d r2 m = (m m )g + m f (v u p 2u) p dt p − f f dt − 2 dt − − 10∇ r2 6πr µ (v u p 2u) − p f − − 6 ∇ 2 t d rp 2 2 dτ (v u 6 u) 6r √πµf ρf − − ∇ dτ. (1.111) − p √t τ Zt0 − The initial conditions are that the sphere is introduced at t = t0 and that there is no disturbance flow prior to this.

34 1. Equations of multiphase ﬂow

By comparing Eqn. (1.111) with the BBO equation, Eqn. (1.109), it is seen that the Stokes drag, the added mass and the Basset history terms have been modified by the inclusion of terms containing the Laplacian of the fluid velocity. These modifications thus include the effects of velocity curvature on the drag force of a particle at low Reynolds numbers, i.e., Faxen relations [Fax22]. Other modifications of the BBO equation than those discussed above can be found in the literature (see, e.g., [Aut83, Buy66]). In many practical cases, the most important terms on the right side of the BBO equation or of Eqn. (1.111) are the gravity term and the Stokes drag term. Especially for rapidly accelerating or oscillating flows also the other terms may become important, however. In order to select an appropriate form of the equation, it is thus important to estimate the magnitude of all the force terms in each particular case. In most applications, the Basset history term is ignored either as insignificant, or due to excessive numerical complications brought about by the integral included in this term. It can be shown that for a laminar time dependent flow (and for the mean field of a turbulent flow), Eqn. (1.111) is valid provided that [MR83]

r r W r2 U p 1, p 1 and p 1, (1.112) L νf νf L where L is the characteristic length scale of the system, W is the characteristic relative velocity (v u) and U/L is the scale of fluid velocity gradient − for undisturbed flow. For a turbulent flow there is no single set of scales but a continuous spectrum of velocity and length scales. The large scale energetic motions may be characterized by the integral turbulent length scale Lt and by the 2 rms velocity urms = δu . The dissipative small scale motions are described by the Kolmogorovp microscales of length and velocity ν3 1/4 η = f and v = (ν )1/4 , (1.113) k k f respectively. Here, denotes the dissipation rate of turbulent energy. These two limiting scales are related by

ηk 1/2 vk 3/2 = Reλ− and = Reλ− , (1.114) Lt O urms O n o n o where Reλ is the Reynolds number deﬁned by the Taylor microscale λ, i.e. Reλ = urmsλ/νf [TL72]. The steepest velocity gradients are found at the dissipative scales. The appropriate scale of the velocity gradient is thus given by vk/ηk, or equivalently by urms/λ. For a turbulent ﬁeld the conditions of validity of Eqn. (1.111) are thus given by

r r W r W r2 v r2 p 1, p = p 1 and p k = p 1 . (1.115) η ν O η v ν η O η2 k f k k f k ( k ) 35 1. Equations of multiphase ﬂow

These conditions are usually much more restrictive than the conditions given by Eqn. (1.112) for the mean velocity field of a turbulent flow, and for a laminar flow. Provided that the conditions given by Eqn. (1.115) are fulfilled, the path of an individual particle in a turbulent flow can in principle be solved using the preferred form of Eqn. (1.111) (or of Eqn. (1.109)). However, this straightforward approach is not feasible, in general, since the full turbulent flow field u of the carrier fluid is not known. Instead, we may assume that the mean (time averaged) velocity u¯ and the necessary statistical properties of the turbulence are known (as given by measurements or by an appropriate turbulence model). The particle tracking method then involves first solving the mean velocity of the particle v¯ at a given instant of time t using Eqn. (1.111) where the turbulent fluid velocity field u is replaced by the mean velocity u¯ = u δu (and δu is the turbulent velocity fluctuation). The dis- − placement of the particle due to the mean flow during a short time interval ∆t is given by ∆y¯ = v¯∆t. An additional stochastic displacement δy due to turbulent dispersion must then be added to yield the total displacement ∆y¯ + δy of the particle at time interval ∆t. The path of the particle during a finite time is found by iterating this process through subseqent short time intervals. The pathline of an individual particle thus consists of a smooth contribution from the mean flow, a possible drift due to gravity or other external forces, and of a twisted random walk contribution due to turbulence. When applied to a large number of particles, this approach leads to a convection-diffusion type of behaviour of the dispersed phase. If conditions of one-way coupling prevail the location of the particle can be found by the described integration process with the knowledge of the existing fluid field, available by preceding solution of fluid phase. For conditions of two-way coupling this field is known just after solving the particle trajectory changing the process inherently to coupled and nonlinear. Therefore, alternating iterative solution of fluid and dispersed phases until convergence is necessary. Calculation of the dispersive displacement yet requires additional modeling to find the propability distribution for the random variable δy. This can be done making use of Eqn. (1.111) and assuming only the necessary spectral characteristics of the fluid turbulence to be known. Specific particle dispersion models are discussed further in the next section.

1.5.2 Particle dispersion The most important property in characterizing the response of particles to the fluid flow is their hydrodynamic response time tp. For a rigid sphere in Stokes flow (Rep < 1), tp is given as 2 ρpdp tp = , (1.116) 18µf

36 1. Equations of multiphase ﬂow

where ρp is the density of particles, dp the diameter and µf the fluid dynamic viscosity. This characteristic time is related to the particle inertia and form drag. Above the Stokes regime, tp depends on the particle Reynolds number Rep defined in terms of the relative velocity between the particle and the fluid ufp ρf dp(u v) ρf dpufp Rep = − = . (1.117) µf µf

Thus, for Rep > 1 tp is deﬁned as

4ρpdp tp = , (1.118) 3ρf CDufp where CD = CD(Rep) is the drag coefficient of the particle. As the particle is transported by the mean flow and dispersed by the turbulence, scales for both of them are required. The mean flow can be scaled by a mean characteristic velocity U and a characteristic length scale L. The Stokes number, defined by t U St = p , (1.119) L gives the ratio of the hydrodynamic response time to the time scale of the mean flow. It thus indicates how well the particle can respond to the mean flow. To describe the dispersion by the turbulence both Lagrangian and Eu- lerian scales of turbulence are required as the dispersion of small neutrally buoyant particles is governed by the Lagrangian scales and the dispersion of large heavy particles is dominated by the Eulerian scales. For Eulerian integral time scales, a scale related to the frame moving with the mean flow TmE and a scale determined with a fixed frame TfE exist. The former can be measured and the latter can then be calculated but no device exist for the direct measurement of Lagrangian integral scales TL. The response of the particles to turbulence is commonly expressed by a Stokes number in which TmE is used as the fluid time scale t St = p . (1.120) TmE The importance of the Stokes number and the above parameters can be demonstrated by expressing the equation of motion of a single particle Eqn. (1.109) in non-dimensional form

dv+ (u+ v+)f γδ = − i3 . (1.121) dt+ St − St In Eqn. (1.121), for the purpose of simplicity, only the ﬁrst and the ﬁfth terms corresponding to form drag and buoyancy have been retained. In addition, the parameter f expresses the ratio of the form drag to the Stokes

37 1. Equations of multiphase flow drag and the gravitational field has been assumed to point in the negative x direction. As characteristic values, uδ 2 for velocity, T for time 3 h i mE and uδ 2 T for length have been used. h i mE p Particlesp of different material than the surrounding fluid do not follow equivalent paths with the fluid. This means that the knowledge of the Lagrangian temporal correlation of velocities for the fluid is not adequate but the corresponding fluid-particle correlation (tensor) is required. Here, the possibilities for theoretical treatment are limited on only a few special cases. If the properties of the particles are close to that of the fluid, it can be assumed that a given particle stays inside the same turbulent eddy for the entire life time of the eddy. This condition can also be stated in the form that the particles essentially stay in the highly correlated part of the flow and, consequently, their dispersion follows that of the fluid. For this case the motion of the particle can be approximated by the linearized form of the equation of motion of the particle, i.e., the Tchen’s solution [Tch47], discussed in section 1.6.5. A new theoretical problem, called ’the preferential concentration of particles’ [EF94], is raised with these almost neutrally buoyant particles. That indicates a condition where the particles are not randomly distributed in the fluid but become concentrated on certain areas of the turbulent structures. If, instead, particles have a notable relative speed with respect to the surrounding fluid, the particle will leave the eddy before the eddy decays. Such condition is typically found in gas-particle systems and in the presence of strong external force field. As a consequence of this ’effect of crossing trajectories’, the particles rapidly loose their velocity correlation to the fluid. This effect can also lead to strongly unisotropic dispersion. On the other hand, in many engineering problems of particulate flows the dispersion is mainly limited by this effect and, consequently, the difficulties related to more detailed description of turbulence and preferential concentration of particles can be avoided. The effect of crossing trajectories is further discussed in section 1.6.5 based on the work of Csanady [Csa63b, Csa63a]. The theoretical treatment of particle dispersion rests, in general, on the assumption of stationary and homogeneous turbulence field. The fundamental work concerning the statistical diffusion of fluid points was made by Taylor [Tay21] and later generalized by Batchelor [Bat53]. This formalism can be directly applied to the dispersion of particles as well. When Brownian motion is neglected as compared to turbulent contribution, the random continuous motion of a single particle is defined by the mean square displacement tensor over an ensemble of realizations of the system

t t0 2 2 yL,i yL,j(t) = vL,(i) vL,(j) RLp,ij(τ)+ RLp,ji(τ) dτ dt0 h i h i h i 0 0 { } q Z Z 38 1. Equations of multiphase ﬂow

t 2 2 = vL,(i) vL,(j) (t τ) RLp,ij(τ) + RLp,ji(τ) dτ, (1.122) h i h i 0 − { } q Z where yL stands for the location of the particle in reference to the coordinate system moving with the mean particle velocity (Lagrangian coordinate system) and vL the instantaneous Lagrangian velocity of the particle. Here, parentheses around the subscript are used to note that the Einstein summation convention must not be used. The latter equality sign holds for the stationary and homogenous ﬁeld under study. Further, the Lagrangian temporal correlation tensor for particle velocities is deﬁned as

vL,i(0) vL,j(τ) RLp,ij(τ) = h i . (1.123) v2 v2 h L,(i)i h L,(j)i q By denoting the symmetric part of RLp,ij(τ) as

1 RS (τ) = R (τ) + R (τ) , (1.124) Lp,ij 2{ Lp,ij Lp,ji } the mean square displacement tensor can be expressed by

t 2 2 S yL,i yL,j(t) = 2 vL,(i) vL,(j) (t τ)RLp,ij(τ) dτ . (1.125) h i h i h i 0 − q Z Since the Lagrangian temporal correlation tensor RLp,ij(τ) and the power spectral density ELp,ij(ω) are Fourier transform pairs of each other [TL72]

R (τ) = ∞E (ω) exp( iωτ)dω , (1.126) Lp,ij ∞ Lp,ij − Z− where ω corresponds to the frequency of a temporal harmonic oscillation into which the motion of the particle is decomposed, it is possible to transform Eqn. (1.125) into the form

t 2 2 ∞ S yL,i yL,j(t) = 2 vL,(i) vL,(j) (t τ) ELp,ij(ω) cos(ωτ) dω dτ h i h i h i 0 − 0 q Z Z 2 2 ∞ S 2(1 cos ωt) = vL,(i) vL,(j) ELp,ij(ω) − 2 dω . (1.127) h i h i 0 ω q Z To calculate y y (t) by Eqn. (1.125) or by Eqn. (1.127) the relationship h L,i L,j i either between RLf,ij(τ) and RLp,ij(τ) or between ELf,ij(ω) and ELp,ij(ω) is needed. The direct relation between the Lagrangian temporal correlation tensors of the ﬂuid and of the particle is very complicated, in general. Some approximate solutions are, however, available for the relation between the two power spectral densities [TL72].

39 1. Equations of multiphase ﬂow

The classical dispersion tensor for a cloud of particles can now be defined as 1 d D = y y (t) p,ij 2 dt {h L,i L,j i} 2 2 ∞ S sin ωt = vL,(i) vL,(j) ELp,ij(ω) dω . (1.128) h i h i 0 ω q Z The above analysis has been carried out with ensemble averages as is appropriate in the general theory of random processes. However, as the analysis concerns only stationary and homogeneous fields the random fluctuations are ergodic [TL72] and, consequently, time averages become identical to ensemble averages for long integration times. Since it is always possible to use a Cartesian frame of reference where S S S S ELf,ij(ω) and ELp,ij(ω) (and therefore RLf,ij(ω) and RLp,ij(ω)) are simultaneously diagonal, the real 3-D problem can be reduced to a combination of three one-dimensional problems without any further loss of generality. Therefore, the following treatment utilizes this finding by concentrating only on a single index for which, e.g., the above correlation tensors and power spectral densities are denoted as RLf (ω), RLp(ω), ELf (ω) and ELp(ω). Thus, in order to calculate the mean square displacement, the rms fluctuating velocities of particles and the Lagrangian temporal correlation tensor must be known. Of these two quantities, the latter poses the main difficulty. Even for fluid particles this correlation tensor is not generally known as there is no device available that would be capable of following the tracked particle and directly measuring it’s velocity. Measurement systems commonly utilize fixed Eulerian coordinate system, and consequently, produce data from which Eulerian correlations of velocities can be determined. Unfortunately, no simple relation between these two types of correlation tensor exist, except in the case of stationary homogeneous fields.

Eddy interaction model The ’eddy interaction model’ is a customary approach to calculate (or more accurately, numerically simulate) turbulent dispersion in the Lagrangian approach. It’s original version is based on the discrete eddy velocity specifi- cation and constant characteristic scales of eddies throughout the flow field [HHD71, BH79]. Later on this approach was extended to complex flows by defining the eddy scales from the turbulent statistics available from the turbulence model used within the carrier fluid. A large number of articles on this approach exist, e.g., [GI81a, Fae83, WBAS84, GHN89, SAW92]. It is also utilized in several commercial flow simulation codes. In the eddy interaction model, the particle motion is determined by the interaction of a particle with a succession of turbulent eddies in which the velocity is kept constant within a finite volume, characterized by the length and time scales

40 1. Equations of multiphase ﬂow of the eddy. The attraction of this model lies in its conceptual simplicity and in that the only statistics required is given by the characteristic scales of velocity, time and length. In the alternative models the forms of either the temporal Lagrangian or the spatial Eulerian velocity autocorrelation functions are required [BDG90, BB93, LFA93].

particle path v y 1 f l uid eddy path u 2 L e x 0 , y 0 x 1

t 0 t 1

Figure 1.3: Illustration of the eddy interaction model and related parameters. An illustration of the eddy interaction model and related properties is shown in Fig. 1.3. At initial time t0, the particle resides at the center of the fluid eddy located at x0, i.e., y0 = x0. At some later time t1, the fluid eddy has traveled to a new location x1 = x0 + t1u = x0 + t1( u + δu). Because of the velocity difference between the fluid eddy and the particle, their paths are not identical but the particle will be found at y1 = x0 + t1(u + v). The distance between the center of the eddy and the particle is therefore (y x ) = t v. The particle remains under the influence of the current eddy 1− 1 1 of size Le until either the life time of the turbulent eddy Te is exceeded, or it travels out of the eddy due to the velocity difference within an interaction time Ti (the effect of crossing trajectories). When either of these conditions is satisfied, a new eddy is generated and the process is repeated. For Stokesian drag and negligible contributions from fluid pressure gradient, added mass and Basset history terms, it is possible to derive analytical expressions for the particle velocity and position at the end of interaction with an eddy. For that purpose it is assumed that the particle enters the eddy at x0 with velocity of v0. The fluid velocity Ue within the eddy is constant over the interaction time Ti. The equation of motion of the particle is now given by d U v(t) (v(t)) = e − , (1.129) dt tp

41 1. Equations of multiphase ﬂow

where, as before, tp is the particle hydrodynamic response (relaxation) time, Eqn. (1.116). Solving this equation gives the updated particle velocity after the interaction with the eddy as

v = U (U v ) exp( T /t ). (1.130) 1 e − e − 0 − i p The updated position of the particle is then given by

t x(t) = x0 + v(t0)dt0 (1.131) Z0 = (x + T U ) (U v )t (1 exp( T /t )) . (1.132) 0 i e − e − 0 p − − i p The particle crossing time T is defined by the equation x(T ) x T U = c | c − 0− c e | Le. The solution is given by L T = t log(1 e ) , (1.133) c − p − V t | fp | p where V = U v is the fluid-particle relative velocity. This expression is fp e − 0 valid only if Le/(Vfptp) < 1, and then Ti is set equal to Tc. If this inequality does not hold, i.e., the particle stays inside the eddy, the interaction time Ti is set equal to the eddy life time Te. Because the interaction time can be determined prior to the interaction, only one integration per eddy is required. In the case of small relaxation time tp, the particles are almost always captured by the eddies. The interaction time will thus be equal to the eddy life time and the particle velocity will quickly approach the fluid velocity. Then, in the context of eddy interaction model, the dispersion of particles will be similar to the dispersion of fluid points. For large relaxation times the interaction time is more often set by the particle crossing time and, consequently, fluid-particle interaction will be almost independent of the eddy life time. As emphasized above the key problem of the eddy interaction model is to find the appropriate scales for the velocity, time and length of the eddies. In general, these scales are random variables and should be given by the statistical properties of fluid turbulence available from the applied turbulence model or from experimental data. Thus, the components of, e.g, eddy velocity can be randomly sampled from a Gaussian velocity distribution with a zero mean and standard deviation uδ 2 = 2 k, where k is the h i 3 kinetic energy of fluid turbulence. p q The eddy life time (life time distribution) can be related to the La- grangian integral time scale of fluid turbulence, given by

∞ TLf = RLf(τ) dτ, (1.134) Z0 42 1. Equations of multiphase ﬂow where u (t) u (t + τ) R (t, τ) = h L L i (1.135) Lf u2 (t) h L i is the Lagrangian temporal correlation tensor of ﬂuid. In a stationary turbulence RLf is independent of time, i.e.,RLf (t, τ) = RLf (τ). Assuming that eddy life times te are distributed according to a probability distribution function f(te), we can write the Lagrangian temporal correlation in the form [WS92]

τ∞(te τ)f(te) dte RLf (τ) = − . (1.136) R 0∞ tef(te) dte The most simple choice for the eddyR life time distribution is the delta function f(t ) = δ(t T ), which corresponds to a constant eddy life time T e e − e e [HHD71, BH79, JHW80, WAW82, GI81a, GHN89]. The Lagrangian temporal correlation then assumes the form

1 τ/T , τ T R (τ) = − e ≤ e . (1.137) Lf 0 , τ > T e Using Eqn. 1.134, we then obtain the required correlation within the constant life time scheme as. Te = 2TLf . (1.138) Another assumption that has been used in the context of eddy interaction simulations [KR89] is the exponential distribution of life times, i.e., f(t ) = exp( t /T )/T , where T is the mean life time. In this case, the e − e m m m Lagrangian temporal correlation is given by

R (τ) = exp ( τ/T ). (1.139) Lf − m Using Eqn. 1.134 we now obtain

Tm = TLf, (1.140) which again correlates the eddy life time distribution with the appropriate turbulent time scale within the exponential distribution scheme. Notice, that for the eddy interaction model it has been shown [GJ96], that the mean eddy life time Tm is subject to the general constraint that T T 2T . The two simple schemes discussed above thus corre- Lf ≤ m ≤ Lf spond to the limiting cases allowed by this inequality. As the number of samples that are required for adequate prediction of particle dispersion in eddy interaction models depends on the mean eddy life time and, therefore, on the selected life time distribution, the constant eddy life time scheme requires the lowest, and the exponential scheme the highest number of samples of all the admissible distributions. Furthermore, to enable a realistic

43 1. Equations of multiphase ﬂow

statistical configuration of particles at the initial time t = t0, i.e., at the moment of particle release, the life time of the first eddy in constant life time scheme has to be randomly distributed according to a uniform distribution f0(te) = 1/Te, 0 < τ < Te [GJ96]. Within the exponential scheme instead, a realistic configuration of particles at the initial time is automatically generated. For large slip velocity Vfp, the eddy size distribution can be related to the Eulerian integral size scale of fluid turbulence (similarly with the eddy life time discussed above). This size scale is defined by

∞ LEf = REf (λ, 0) dλ, (1.141) Z0 where u(x, t)u(x + λ, t + τ) R (λ, τ) = h i (1.142) Ef u2(x,t) h i is the Eulerian fluid velocity autocorrelation function. (In a general case, a much more complex Lagrangian correlation of fluid along the particle path and a corresponding Lagrangian scale should be used [Csa63b, Csa63a].) Analogously with the eddy life time distribution, this autocorrelation function can be expressed in terms of eddy size distribution g(le). Assuming constant eddy length L [HHD71] corresponds to g(l ) = δ(l L ) and, in e e e − e the case of stationary turbulence, gives the autocorrelation function of the form 1 λ/L , λ L R (λ) = − e ≤ e . (1.143) Ef 0 , λ > L e Using Eqn. (1.141) then gives the result Le = 2LEf . In general, the length scale can be taken to be a random variable with a given distribution function. An exponential probability distribution, e.g., gives results completely analogous to Eqns. (1.139) and (1.140) [BM90]. Another possibility for finding the relevant time and length scales is given by the widely used model of Gosman and Ioannides [GI81a]. Within that model it is assumed that the eddy time and length scales are related to the ’dissipation scales’ L and T, given by

k3/2 k L = C3/4 and T = 3/2 C3/4 , (1.144) µ µ p where k is the kinetic energy of turbulence, its rate of dissipation and Cµ 2 is the constant that appears in deﬁnition of eddy viscosity µT = ρCµk / in the standard k turbulence model. − Notice, however, that even though we may assume the eddy life time Te and size Le to be proportional to the corresponding statistical scales given above, the actual relation depends on the details of the underlying statistical model.

44 1. Equations of multiphase ﬂow

1.6 Practical closure approaches

In this section we shall give more detailed examples of constitutive relations for selected systems. The constitutive relations given here finally render the general flow equations for a given multiphase system derived in section 1.2.2 in a closed form amenable for numerical solution (see Chapter 2 below). The final equations derived here may not however be generally valid for all such systems and should merely be taken as examples of plausible constitutive models that may be used for that particular type of systems. One should also bear in mind that the existing models in particular for turbulence in multiphase flows still lack generality and can be considered inadequate to some extent. For the sake of reliability of the model it is thus essential that the values of various material parameters, that are left unspecified in the constitutive relations, are determined by independent measurements in conditions that closely resemble those of the actual application. It is also essential that the numerical solution is verified experimentally.

1.6.1 Dilute liquid-particle suspension As a starting point, we use the Eqns. (1.37) through (1.41) derived in section 1.2.2. We thus have eight equations for the eight unknowns which can be taken to be the volume fraction of the fluid φ, fluid pressurep ˜f and the three components of both the velocities u¯f and u¯d. It remains to specify the constitutive relations for the viscous stress tensor of the fluid τ , the total h f i stress tensor of the particulate phase σ , the momentum transfer integral h di D, and the turbulent stresses τδd and τδf . The constitutive relation for the viscous stress tensor of the fluid τ can h f i be derived simply by performing the volume averaging of the mesoscopic tensor τ = µ (( u ) + ( u )t) and using Eqn. (1.14). We define the f f ∇ f ∇ f averaged surface velocity of the fluid U¯ S by 1 1 u nˆ dA = U¯ nˆ dA = U¯ φ, (1.145) V f f S V f − S∇ ZAf ZAf and postulate that U¯ = bu¯ (1 b)u¯ , where b = b(φ) is a free parameter S d − − f (the ’mobility’ of the dispersed phase) which aquires values between 0 and 1. It is then easy to see that the viscous stress tensor of the fluid can be given by

τ = φτ˜ h f i f = φµ [( u¯ ) + ( u¯ )t] (1.146) f ∇ f ∇ f bµ [( φ)(u¯ u¯ ) + (u¯ u¯ )( φ)]. − f ∇ d − f d − f ∇ For a very dilute liquid-particle suspension where the collisions between particles can be ignored, the stress state of the particles is determined by the

45 1. Equations of multiphase flow hydrodynamic forces exerted on the surface of the particles by the surrounding fluid. If the velocity difference between the particles and the fluid is not very high, we may approximate

σ = (1 φ)˜σ (1 φ)˜p 11. (1.147) h di − − d ≈ − − f The term σ p˜ φ on the right side of Eqn. (1.41) is now reduced to ∇ · h di − f ∇ (1 φ) p˜ . For a dense suspension instead, the tensor σ is contributed − − ∇ f h di by particle particle collisions and may have a very complicated form which we do not consider here (see [Hwa89] and references therein). According to Eqn. (1.36), the transfer integral D is the force per unit volume acting on particles due to fluctuations of fluid pressure and due to viscous stresses. From the standard fluid mechanics we know that specific hydrodynamic forces act on particles that move through the fluid with constant velocity, with acceleration or with superimposed linear motion and rotation. In principle, the forces acting on such a particle (with low particle Reynolds number) are specified by the BBO equation, Eqn. (1.109) or Eqn. (1.111). Thus, the transfer integral should contain a contribution from all the force terms that appear on the right side of that equation - except of the terms that arise from gravitational force and from the fluid pressure gradient, since these effects are already included in Eqns. (1.39) and (1.41)! For a dilute suspension where particles can be considered independent of each other, the terms in D corresponding to various terms in the selected version of the BBO equation can be derived in a rather straightforward manner. For example, neglegting the effects of fluid velocity gradient, and taking into account only the Stokes drag force and the added mass term (which arises since accelerating a particle immersed in a fluid accelerates an amount of fluid around the particle as well), we may write

∂ D = B(u¯ u¯ ) + C ( (u¯ u¯ ) + (u¯ u¯ ) (u¯ u¯ ) , (1.148) d − f ∂t d − f d − f · ∇ d − f h i where B and C are unknown but perhaps measurable parameters. For a very dilute suspension of spherical particles of radius rp and for low particle Reynolds numbers, Eqn. (1.109) suggest that

9µf B = 2 (1 φ) (1.149) 2rp − 1 C = ρ˜ (1 φ). (1.150) 2 f − Many other interaction mechanisms than those included in Eqn. (1.148) may be crucial in practical flows of fluid-particle suspensions [Dre83]. For instance if a particle is rotating, moving in the presence of a strong velocity gradient of the fluid or, especially, if the particle is moving near a solid wall, it may experience a ’lift’ or ’side’ force which is perpendicular to its direction

46 1. Equations of multiphase flow of relative motion with respect to the fluid. A particle moving in a fluid may also experience hydrodynamic forces which depend on the previous history of its motion. This effect is taken into account in the BBO equation through the Basset history term which may become important for instance in the case of fast oscillatory motions (see [Soo90] and references therein). The tensors τ and τ contain the momentum transfer due to turbu- h δf i h δdi lent and ”pseudo-turbulent” fluctuations of the two phases. Various models have been proposed for these quantities which are analogous to the ordinary Reynold’s stresses in single phase flows. The constitutive relations suggested by Drew and Lahey are [DLJ92]

τ = φpT 11 + 2µT Π + 2µT Π h δf i − f ﬀ f fd d +(1 φ)ρ [a u¯ u¯ 211 + b (u¯ u¯ )(u¯ u¯ )] (1.151) − f f | f − d| f f − d f − d

τ = (1 φ)pT 11 + 2µT Π + 2µT Π h δdi − − d dd d df f +(1 φ)ρ [a u¯ u¯ 211 + b (u¯ u¯ )(u¯ u¯ )],(1.152) − d d| f − d| d f − d f − d 1 t where Πα = 2 [( u¯α) + ( u¯α) ] for α = f, d. The turbulent pressures T ∇ T∇ pα , the eddy viscosities µαβ and the coefficients aα and bα are still to be correlated with appropriate variables that characterise the turbulent and pseudoturbulent motion of the phases. This might be accomplished through experiments or through additional turbulence modelling [CTC96]. Here, we shall consider a simple generalization of the ordinary k model for − onephase flows. In this illustrative multiphase version or the k ε model we ignore all − the other terms on the right side of Eqns. (1.151) and (1.152) than those T T proportional to the eddy viscosities µff and µdd. We also assume that the eddy viscosity hypothesis holds for each individual phase. Thus we can define the eddy viscosity for phase α in analogy with single phase flows as

2 T kα µαα = Cµρ˜α , (1.153) εα where kα and εα are the turbulent kinetic energy and the dissipation for phase α, respectively, and Cµ is an empirical constant. The transport equations for kα and εα are postulated to have the form

T ∂ µαα (φαρ˜αkα) + (φαρ˜αu¯αkα) [φα(µα + ) kα] ∂t ∇ · − ∇ · σk ∇ = φαSkα + Γkα (1.154) T ∂ µαα (φαρ˜αεα) + (φαρ˜αu¯αεα) [φα(µα + ) εα] ∂t ∇ · − ∇ · σε ∇ = φαSεα + Γεα. (1.155)

47 1. Equations of multiphase ﬂow

Analogously with single phase ﬂows the source terms are divided into pro- duction and dissipation terms as follows

S = P ρ˜ ε (1.156) kα α − α α εα Sεα = (C1εPα C2ερ˜αεα) (1.157) kα − P = µT u¯ [ u¯ + ( u¯ )t]. (1.158) α αα∇ α · ∇ α ∇ α

Interphasial turbulence exchange terms Γkα, Γεα must be deﬁned separately for each case. Assuming that the turbulence quantities are equal for both phases, i.e., that k k and ε = ε for all α = f, d, and summing the above α ≡ α transport equations, we get

∂ µT (ρk) + [ρu¯k (µ + k)] = Sk (1.159) ∂t ∇ · − σk ∇ ∂ µT (ρε) + [ρu¯ε (µ + ε)] = Sε, (1.160) ∂t ∇ · − σε ∇ where

ρ = φαρ˜α (1.161) α X 1 u¯ = ( φ ρ˜ u¯ ) (1.162) ρ α α α α X µ = φαµα (1.163) α X T T µ = φαµαα (1.164) α X Sk = φαSkα (1.165) α X Sε = φαSεα. (1.166) α X 1.6.2 Flow in a porous medium Most porous materials of practical interest consist either of particles packed in a more or less distordered manner or of a consolidated irregular porous structure of some kind. Examples of such materials are numerous: sand, soil, fractured rock, ceramics, sponge, paper etc. Many important processes found in geophysics or in various industrial applications involve flow of fluid through a porous medium. In some cases, such as in slow transport of ground water through an aquifer, the porous material can be considered rigid so that the structure of the solid matrix is not significantly deformed during the process. The basic equation for such a flow is given by the famous Darcy’s law, which was originally inferred from purely empirical results for a

48 1. Equations of multiphase flow stationary creeping flow of Newtonian liquid through a homogeneous column of sand [Bea72]. With processes such as removal of water from a sponge by squeezing it, the porous structure appears soft and may thus be extensively deformed by external forces and by hydrodynamical forces exerted on the solid matrix by the fluid flow. In this section, we shall utilize Darcy’s experimental formula in the context of the multiphase flow theory and derive the governing equations for time dependent creeping flow of Newtonian liquid through a soft porous medium. Formally, we treate the system of the highly deformable solid matrix and the liquid flowing through the interstities of the matrix as a binary mixture of two fluids. We assume again that both phases are incompressible, that there is no mass transfer between the two phases and that surface tension between the solid material and the liquid is negligible. The situation is thus analogous with the liquid-particle suspension discussed in section (1.2.2). However, instead of a contiunous liquid phase and a dispersed particle phase we now have two continuous phases. Replacing the subscript ’d’ for ’dispersed’ phase by ’s’ for ’solid’ phase, the Eqns. (1.37) through (1.41) are thus formally valid also for the present system. Several simplifications as compared to the liquid-particle suspension can however be made in this case. Assuming creeping flow indicates that the inertial terms that appear on the left side of Eqns. (1.39) and (1.41) can be neglected. Furthermore, the pseudoturbulent stress term τ vanishes for the solid ∇·h δdi phase and is expected to be very small also for the fluid phase in this flow regime. According to Darcy’s early experiments and to innumerable later experiments, the dominant interaction mechanism in a flow through porous medium is viscous drag. The results of these experiments, as summarized by the Darcy’s law, indicate that the momentum transfer integral D should be written in a form µ D = (u¯ u¯ ). (1.167) − k f − s Here, k = k(φ) is the permeability of the porous material which remains to be measured. Several experimental correlations for k has been reported in literature for different types of porous media [Bea72]. Perhaps the most common formula which can be derived analytically for simplified capillary models of porous materials and which at least qualitatively grasps the correct behaviour for many materials, is the Kozeny-Carman relation 1 φ k = . (1.168) cS2 (1 φ)2 0 − Here, S0 is the specific pore surface area and c is the dimensionless Kozeny constant which aquires values between 2 and 10, in practice. (Notice that due to the conventions used here, Eqn. (1.168) differs from its more usual form where φ3 instead of φ appears in the nominator, see Eqn. (1.177) below.) Furthermore, if the porosity φ is not too close to unity, the viscous

49 1. Equations of multiphase ﬂow shear stress term τ is small as compared to the viscous drag term ∇ · h f i and can be neglected. Taking gravitation to be the only body force, the equations for a ﬂow of liquid in a deformable porous medium can thus be written in a form

∂ φ + (φu¯ ) = 0 (1.169) ∂t ∇ · f ∂ (1 φ) + [(1 φ)u¯ ] = 0 (1.170) ∂t − ∇ · − s

µ φ p˜ = (u¯ u¯ ) + φρ˜ g (1.171) ∇ f − k f − s f µ σ = + (u¯ u¯ ) p˜ φ + (1 φ)˜ρ g. (1.172) −∇ · h si k f − s − f ∇ − s

Adding Eqns. (1.169) and (1.170) and Eqns. (1.171) and (1.172) we arrive at the mixture equations

q = 0 (1.173) ∇ · h i T = ρ g, (1.174) ∇ · h i −h i where q = φu¯ + (1 φ)u¯ is the volume flux, T = φp˜ 11 + σ is the h i f − s h i − f h si total stress, and ρ = φρ˜ + (1 φ)˜ρ is the density of the mixture. h i f − s For linearly elastic materials, the stress tensor σ is readily given as a h si function of local strain by Hooke’s law. For viscoelastic materials instead, σ may depend both on the strain and on the rate of strain (i.e. on u¯ ). h si s Since the solid phase is actually not a fluid in an ordinary sense, a finite stress implies finite strain on the solid matrix. It follows that the velocity of the solid phase can be non-zero only in a transient state. In a stationary state (and in the case of rigid porous material) we have u¯s = 0. The porosity φ is then independent of time, and the flow equations are reduced to

q = 0 (1.175) ∇ · f k¯ q = ( p˜ + ρ˜ g), (1.176) f −µ ∇ f f and one of Eqns. (1.172) or (1.174). Here qf = φu¯f is the volume ﬂux of the ﬂuid (the ’seepage’ velocity), and

1 φ3 k¯ = . (1.177) cS2 (1 φ)2 0 − Eqn. (1.176) is the Darcy’s formula in its conventional form.

50 1. Equations of multiphase ﬂow

1.6.3 Dense gas-solid suspensions The behaviour of solid particles in a dense gas-solid suspension is strongly affected by the binary interparticle collisions. The kinetic theory of granular flow is derived for this special case of twophase flow in analogy with the kinetic theory of dense gases. A conservation equation (the Bolzmann equation) for the particulate phase is formulated in terms of the single particle velocity distribution function f 1(x, u, t)

∂ 1 ∂ 1 ∂ 1 ∂ 1 f + (uif ) + (Fif ) = f (1.178) ∂t ∂xi ∂ui ∂t !coll where F is the external force per unit of mass acting on a sphere and the right side describes the rate of change of the distribution function due to particle collisions. In the kinetic theory of granular ﬂow the ensemble-average (1.59) of a function ψ(u) is deﬁned by

1 ψ(u) = ψ(u)f 1(x, u, t)du (1.179) h i n p Z where np is the number of particles per unit volume. A transport equation for ψ(u) can be derived from Eqn. (1.178) by multiplying it by ψ(u) and h i integrating it over the velocity domain [CC70]:

∂ ∂ (np ψ ) + (np ψui ) (1.180) ∂t h i ∂xi h i ∂ψ ∂ψ ∂ψ np + ui + Fi = C(ψ), − * ∂t + * ∂xi + * ∂ui +! where the collisional rate of change for ψ is deﬁned by ([JR85], cited in [Pei98])

∂ ∂us,j ∂ψ C(ψ) = χ(ψ) θi(ψ) θi (1.181) − ∂xi − ∂xi *∂δuj + where us,j is the mean velocity of the discrete phase. Alternative formulations of C(ψ), where the last term is missing, can be found in literature (e.g. [JS83, DG90]). The source term χ(ψ) describes loss of ψ due to inelastic collisions and θi(ψ) transport of property ψ during collisions. These terms are deﬁned by integrals involving the pair distribution function 2 f (x1, x2, u1, u2, t) and can be calculated analytically. In the derivation, the pair distribution function is given as a product of the single velocity distribution functions and a correction function g0. Kinetic theory yields the continuity equation and the momentum equation of a phase in a form similar with the traditional multiﬂuid equations.

51 1. Equations of multiphase ﬂow

The continuity equation is obtained using ψ = 1 and the momentum equation by using ψ = u. In the resulting equations, the kinetic and collisional contributions of the particulate phase stress are treated together. The isotropic parts are described as a solid pressure and the rest as a shear stress term. In addition to the continuity and the momentum equations, a ﬁeld equation for the particle ﬂuctuating kinetic energy must be solved. The following formulation can be found, e.g., in [Boe97]. The continuity equation without mass transfer can be written in the form ∂ (φ ρ˜ ) + (φ ρ˜ u¯ ) = 0. (1.182) ∂t α α ∇ · α α α Using the notation of total averaged stresses the momentum equations can be written in the form ∂ (φ ρ˜ u¯ ) + (φ ρ˜ u¯ u¯ ) = σ + M + φ F˜ . (1.183) ∂t α α α ∇ · α α α α ∇ · h αi α α α Averaged total stresses are given by

σ = φ τ˜ φ p˜ 11 φ p˜ 11 (1.184) h pi p p − p p − p g σ = φ τ˜ φ p˜ 11, (1.185) h gi g g − g g wherep ˜p is an ’extra stress’ due to collision of particles and 2 τ˜ = 2µ Π + (µ µ ) tr(Π )11, (1.186) α α α α,b − 3 α · α where µα is the shear viscosity and µα,b is the bulk viscosity of phase α and 1 Π = u¯ + ( u¯ )t . (1.187) α 2 ∇ α ∇ α h i Furthermore assuming that the interfacial momentum exchange term consist of bouyancy and viscous drag term, i.e.,

M =p ˜ φ + B(u¯ u¯ ) g g∇ g p − g M = M p − g we get ∂ (φ ρ˜ u¯ ) + (φ ρ˜ u¯ u¯ ) = φ p˜ + (φ τ˜ ) ∂t g g g ∇ · g g g g − g∇ g ∇ · g g +Dg + φgF˜ g (1.188) ∂ (φ ρ˜ u¯ ) + (φ ρ˜ u¯ u¯ ) = φ p˜ + [φ (˜τ p˜ 11)] ∂t p p p ∇ · p p p p − p∇ g ∇ · p p − p D + φ F˜ , (1.189) − g p p where D = B(u¯ u¯ ). (1.190) g p − g 52 1. Equations of multiphase ﬂow

Both the shear viscosity and the solids ’extra stress’ consist of a kinetic part and a collisional part. The extra stress can be written as follows

p˜p =ρ ˜pΘp[1 + 2(1 + e)φpgo], (1.191) where Θp is the granular temperature describing the ﬂuctuating kinetic energy of the solid material 1 Θ = tr δu δu (1.192) p 3 h p pi and e is the coeﬃcient of restitution for particle collisions and go is the radial distribution function given by [DG90]

1/3 1 − 3 φp go = 1 , (1.193) 5" − φp,max ! # where φp,max is the maximum packing of the solids. Several alternative forms of the radial distribution function have been proposed in the literature. The shear viscosity can be written as a sum of the kinetic part and the collisional part as follows

µp = µp,kin + µp,col, (1.194) where [LSJC84] 4 Θp µp,col = φpρ˜pdpgo(1 + e) (1.195) 5 r π and [GBD92]

2 10√πρ˜pdp Θp 4 µp,kin = 1 + goφp(1 + e) (1.196) 96 φp(1+pe)go " 5 # Also for the shear viscosity several alternatives exist, with biggest diﬀerences in the dilute regions. The bulk viscosity expresses the resistance against compression and is given by 4 Θp µp,b = φpρ˜pdpgo(1 + e) . (1.197) 3 r π The solids pressure, the shear viscosity and the bulk viscosity above are all given as functions of the granular temperature Θ. The following transport equation has been derived for the ﬂuctuating kinetic energy thus yielding the granular temperature [DG90]

3 ∂ (φpρ˜pΘp) + (φpρ˜pΘpup) (1.198) 2"∂t ∇ · # = ( p 11 + τ ): u + (k Θ ) γ + φ , − p p ∇ p ∇ · Θ∇ p − Θ Θ 53 1. Equations of multiphase flow where the first term on the right hand side presents the generation by local acceleration of particles, the second term the diffusion of Θ, the third term the dissipation of Θ and the fourth term the exchange between gas and solid phases. Several different closure relations for these different terms have been suggested (see the review in [Boe97]). The diffusion coefficient can be divided as follows [GBD92]

kΘ = kΘ,dilute + kΘ,dense, (1.199) where 2 Θp kΘ,dense = 2φpρ˜pdpg0(1 + e) (1.200) r π and 2 75 ρ˜pdp πΘp 6 kΘ,dilute = 1 + (1 + e)g0φp . (1.201) 192 (1 +pe)g0 " 5 # The dissipation of ﬂuctuating energy can be described as [JS83]

2 2 4 Θp γΘ = 3(1 e )φpρ˜pg0Θp u¯p (1.202) − dp r π − ∇ · ! and the interphase exchange term as [DG90]

φ = 3BΘ . (1.203) Θ − p Due to the time consumption of the solution of an extra ﬁeld equation, an algebraic equation is often used for calculation of the granular temperature. It is based on the assumption that there is a local equilibrium and all terms but the generation and dissipation of granular temperature can be neglected. The resulting algebraic equation is ([SRO93], cited in [Boe97])

K1φptr(Πp) Θp = − (1.204) 2K4φp p 2 2 2 2 (rK1φp) t (Πp) + 4K4φp K2tr (Πp) + 2K3tr(Πp) + r h i, 2K4φp where ρ˜p K1 = 2(1 + e)˜ρpgo + (1.205) φp 4d ρ˜ (1 + e)φ g 2 K = p p p o K (1.206) 2 3√π − 3 3 d ρ˜ √π 8φ g (1 + e) K = p p [1 + 0.4(1 + e)(3e 1)φ g ] + p o (1.207) 3 2 3(3 e) − p o 5√π ( − ) 54 1. Equations of multiphase ﬂow

2 12(1 e )˜ρpgo K4 = − (1.208) dp√π The typical applications of the kinetic theory of granular flow are in bubbling and circulating fluidized beds. In dense flows of this type the inter-phase drag forces require special closure relations that are based on measurements in dense suspensions. For bubbling beds the models are based on the models for packed beds (Ergun equation [Erg52]) and on the measurements of Richardson and Zaki who studied liquid-solid fluidization [RZ54]. For circulating fluidized beds it is often necessary to apply clustering corrections to the drag forces [Boe97]. In bubbling beds the inter-particle distances are short and particle-particle collisions dominate the flow. In very dense regions the inter-particle friction can dominate over the fluctuating motion and special description of the shear stress term is required [Boe97]. In circulating fluidized beds the gas phase turbulence and particle-gas interactions are important and corrections to the drag force and several others of the closure relations above may be necessary [BBS95].

1.6.4 Constitutive equations for the mixture model The equations of the mixture model were derived from the general multiphase equations in Section 1.4. For practical applications, constitutive relations are needed for the diﬀusion velocity and the viscous and turbulent stresses. Some formulations of the constitutive relations are given below.

Diﬀusion velocity

In a liquid-particle suspension, the momentum source term Dp can be written in the form φpFp Dp = , (1.209) Vp where Vp is the particle volume. Neglecting all other eﬀects except the viscous drag and assuming spherical particles, we can write in the Stokes regime Fp = 3πdpµcucp. (1.210) Using Eqns. (1.209) and (1.210) together with (1.102), the expression for the slip velocity is

2 dp(ρp ρm) ∂ ucp = − g (u¯m )u¯m u¯m (1.211) 18µc " − · ∇ − ∂t # and the diﬀusion velocity follows from Eqn. (1.103). For larger particle Reynolds numbers, Eqn. (1.210) must be replaced by a corresponding model for the drag force.

55 1. Equations of multiphase ﬂow

Viscous stress The viscous stress tensor is approximated in the mixture model by an expression analogous to incompressible single phase ﬂow

τ = µ [ u¯ + ( u¯ )t] (1.212) m m ∇ m ∇ m

It should be noted that the apparent viscosity of a suspension µm is not a well deﬁned property of the mixture, but depends on many factors, including the method of measurement. However, it turns out that, at reasonably low concentrations, it can be correlated in a simple way to the concentration. In mixture model applications, the most often used correlation for the mixture viscosity is that according to Ishii and Zuber [IZ79]; see also [IM84] for a summary of the results. The general expression for the mixture viscosity, valid for solid particles as well as bubbles or drops, is given by

∗ 2.5φpmµ φp µm = µc 1 (1.213) − φpm ! where φpm is a concentration for maximum packing. For solid particles φ 0.62. In Eqn. (1.213), µ = 1 for solid particles and pm ≈ ∗

µp + 0.4µc µ∗ = (1.214) µp + µc for bubbles or drops. Numerous other correlations for the viscosity of solid suspensions are presented in the literature. Rutgers [Rut62] presented a review of various empirical formulas for the relative viscosity. One of the correlations with a theoretical foundation is due to Mooney (cited in [Rut62])

µ 2.5φ ln m = p (1.215) µ 1 1.4φ c ! − p

Turbulence In the mixture model the effects of turbulence appear in the mixture momentum equation as part of the general stress term. Additionally, turbulent effects appear in the fluid-particle interaction term and in the fluctuating components of particle velocity, i.e., as a turbulent stress in the particle momentum equation. In turbulent multiphase flow at low loadings, it can be assumed that both the viscous stresses of the carrier phase and the turbulent stresses of the particulate phase are negligible. The effective viscosity of the continuous phase can then be calculated directly from a turbulence model for the continuous phase, e.g., from the k model. There a correction can be applied − 56 1. Equations of multiphase flow to reduce the turbulence intensity. The models for correcting turbulent viscosity are unfortunately very uncertain. In some cases the results with the corrections can be less accurate than those obtained using the standard k − model. Therefore, simulations should be performed without the corrections and with various correction methods. In turbulent flows the fluid particle interaction force should be written in the form D = Bu¯ + D0 (1.216) p − cp p where Dp0 is the fluctuating part of the force that causes particle dispersion. In turbulent flow the term Dp0 should be added to the right hand side of Eqn. (1.100). In the original multiphase equations, the turbulent dispersion terms are included in the momentum equations of dispersed phases, Eqn. (1.28). In the mixture model, the influence of turbulence must be contained in the equation for the diffusion velocity. This implies additional terms in (1.211) due to the turbulent stresses and the fluctuation part of Dp. Instead of developing a model for those terms directly, we adopt a simpler and more intuitive method. The main effect of the slip velocity is the diffusion term introduced in the continuity equation of the dispersed phase. If the turbulent terms are taken into account, other terms will appear in the continuity equation representing the turbulent diffusion. The simplest way is to model these terms as Fickian diffusion. The continuity equation of the particulate phase is then ∂ (φ ρ˜ ) + (φ ρ˜ u¯ ) = (D φ ) (φ ρ¯ u¯ ), (1.217) ∂t p p ∇ · p p p ∇ · mp∇ p − ∇ · p p mp where Dmp is a dispersion (or diffusion) coefficient. One simple way to estimate the dispersion coefficient is to estimate it from the turbulent particle viscosity as follows ([SCQM96])

T µp Dmp = T , (1.218) σp ρ˜p

T where σp is the turbulent particle Schmidt number for which values of, e.g., 0.34 and 0.7 have been suggested. Csanady’s model [Csa63b] for the turbulent diﬀusion takes into account the crossing trajectory eﬀect. It was developed for a special boundary layer application under special assumptions. Csanady’s model is still today the best model available for predicting particle dispersion and has shown to be fairly accurate even for other applications. The following equation is based on Csanady’s work and the k model [PBG86] − 1/2 2 − T u¯cp Dmp = νc 1 + 0.85 , (1.219) 2/k 3 !

57 1. Equations of multiphase flow where 0.85 is an empirical value determined from particle dispersion data. The equation above is based on the assumption that the carrier phase turbulent Schmidt number is equal to one. In addition, the turbulent Schmidt number of the dispersed phase was defined using the carrier phase diffusivity as a reference.

1.6.5 Dispersion models The theoretical treatment of particle dispersion was discussed earlier in Sec- tion 1.5.2. There, this treatment was further supplemented with the most common approach utilized in numerical simulations involving turbulent particle dispersion, i.e., the eddy interaction model. In that context the two extreme conditions of particles almost following the turbulent ﬂuid motions, and heavy particles crossing the eddies were mentioned. In the following, an analytic approach of particle dispersion is given for these two limiting cases. They constitute the only theoretical conditions for which practical numerical models can be compared.

Linearized equation of motion To examine the relation between fluid and particle correlations or spectral densities we need to solve the equation of motion of a particle. To this end we study the BBO-equation (1.109), which is a nonlinear integro-differential equation. For the Stokes regime Rep < 1 the nonlinearity is caused mainly by the pressure gradient term (II). By using the Lumley’s approximation for this term [Lum57] the BBO-equation for the velocity component i can be written as dv du 2 ∂2u L,i b L,i ν L,i dt − dt − 3 f ∂xx ∂ j j 2 ∂u +ab (v u ) + (v u ) L,i L,i − L,i 3a L,j − L,j ∂x j t d 3a dτ vL,i(τ) uL,i(τ) +b { − } dτ + fi = 0 . (1.220) π √t τ r Zt0 − In this equation fi denotes an external field force per unit effective mass (real mass + virtual mass) of the particle. The parameters a and b are given by 3νf 3ρf a = 2 and b = . (1.221) rp 2ρp + ρf Hinze [Hin75] pointed out that the nonlinearity becomes negligible if the term including derivative in the third term of Eqn. (1.220) is very small, i.e., 2 ∂u L,i 1 . (1.222) 3a ∂xj

58 1. Equations of multiphase ﬂow

This requirement is fulfilled if the particle size is small as compared to the Taylor microscale. However, the restrictions of validity of the particle equation of motion in turbulent field, set by Eqns. (1.115), already includes this requirement. It was further realized by Hinze that additional simplification is possible if the latter part of also the second term of Eqn. (1.220) is negligible. It is reasonable to assume that this requirement is also met if the particle size is sufficiently small as compared to the turbulent structures. Implicit in both these simplifying conditions is the assumption that the particle resides in a locally uniform fluid field. In other words, it is restricted to move in the strongly correlated region of the turbulent field (inside an eddy) for a time comparable to the eddy decay time. In what follows we consider an arbitrary coordinate direction i in a coordinate system where the correlation tensors and spectral densities are diagonal (see discussion at the end of Sect. 1.5.2), and omit subscript i. Under the assumptions made above, the linearized BBO-equation for a velocity component can be written as dv du L b L + ab(v u ) dt − dt L − L t d 3a vL(τ) uL(τ) +b dτ { − } dτ + f = 0 . (1.223) π √t τ r Zt0 − By defining the Fourier transforms of the fluid and particle velocity components as

uˆ (ω) = ∞ u (t) exp( iωt) dt L L − Z−∞ vˆ (ω) = ∞ v (t) exp( iωt) dt , (1.224) L L − Z−∞ and by applying them in Eqn. (1.223), leads to simple relationship between the velocities [Cha64]

3ωa 3ωa a + 2 + i ω + 2 vˆL = uˆL . (1.225) n q3ωa o nω q 3ωa o a + 2 + i b + 2 n q o n q o Multiplying each side of this equation by its complex conjugate and applying ensemble averaging on the resulting equation gives ? vˆ (ω)v ˆ (ω) Ω(1) h L L i = , (1.226) ˆuˆ (ω) u? (ω) Ω h L L i (2) where ω Ω = ω 2 + √6 ω 3/2 + 3 ω + √6 ω 1/2 + 1 (1.227) (1) a a a a a ω 2 3/2 1/2 Ω , b = 1 ω + √6 ω + 3 ω + √6 ω + 1 . (2) a b2 a b a a a 59 1. Equations of multiphase ﬂow

The Lagrangian spectral densities (see Eqn. (1.126)) can be given in terms of the Fourier transformed velocities as uˆ (ω)u ˆ? (ω) E (ω) = h L L i (1.228) Lf u2 h Li vˆ (ω)v ˆ? (ω) E (ω) = h L L i . Lp v2 h Li It thus follows that 2 E u Ω(1) Lp = h Li . (1.229) E v2 Ω Lf h Li (2) Consequently, the (diagonal elements of) the mean square displacement and dispersion tensors of the particle, Eqns. (1.127) and (1.128), can be given in terms of variables defined by the turbulent field of the fluid as

∞ Ω(1) 2(1 cos ωt) y2 (t) = u2 E (ω) − dω and (1.230) h L i h Li Ω Lf ω2 Z0 (2) Ω(1) sin ωt D = u2 ∞ E (ω) dω . (1.231) p h Li Ω Lf ω Z0 (2) A number of conclusions concerning Eqns. (1.230) and (1.231) can be made without actually solving the spectral densities or calculating the integrals. For the low and high frequency limits, the amplitude ratio of the phases is seen to behave as

ω Ω(1) a 0 , 1 and (1.232) → Ω(2) →

ω Ω(1) 2 a , b . (1.233) → ∞ Ω(2) →

For the ratio of the eﬀective masses of the phases b, it is observed that in the case of very heavy particles, where ρ /ρ 0, and in the case of equal f p → densities, where ρf = ρp, the amplitude ratio is given by

Ω(1) b 0 , 0 and (1.234) → Ω(2) → Ω(1) b = 1 , = 1 , (1.235) Ω(2) respectively. These results are all physically very plausible. Furthermore, the asymptotic values of the dispersion coeﬃcient for short and long dispersion times are given by

2 π 2 lim Dp = uL t and lim Dp = vL ELp(0) . (1.236) t 0 h i t 2 h i → →∞ 60 1. Equations of multiphase ﬂow

The relative degree of the dispersion of particles and ﬂuid is given by

2 t 2 sin ωt Dp vL 0 RLp(τ) dτ vL 0∞ ω ELp(ω) dω = h 2 i t = h 2 i sin ωt , (1.237) Df u R (τ) dτ u ∞ E (ω) dω h Li R0 Lf h Li R0 ω Lf For short and longR dispersion times, theR limiting behavior of that ratio is given by D v2 D lim p = h Li and lim p = 1 . (1.238) t 0 D u2 t D → f h Li →∞ f The latter limit results since for ω = 0, ELp(0) = ELf (0) and Ω(1) = Ω(2) (see Eqn. (1.232)). These asymptotic expressions for both short and long dispersion times should hold regardless of the detailed form of either the Lagrangian temporal correlation or power spectral density. The equality of the particle dispersion and the fluid diffusion at long dispersion times is a consequence of the assumptions made while lineariz- ing the particle equation of motion. As stated above, these assumptions indicate that the particle moves in the strongly correlated area of the fluid eddy, since it’s motion has been related to the surrounding fluid through the Lagrangian correlation coefficient. This is plausible if the particle is very small as compared to the fluid eddy or if the densities of the phases are close to each other. Considerably heavier or lighter particles can not follow the fluid motion. However, if the particles are very much smaller than the fluid eddies, e.g. dust particles in the athmosphere, the assumptions may still be valid.

Crossing trajectories In general, a heavy particle can have a significant mean velocity relative to the fluid, while a particle with a density comparable with that of the fluid tends to follow the motion of the fluid more closely. Therefore, a heavy particle continuously changes it’s fluid neighborhood and drifts away from the fluid eddy in a time scale that is small as compared with the eddy decay time. The velocity correlation of heavy particles with the surrounding fluid thus decreases rapidly and their dispersion rate is low. This phenomenon is known as the effect of crossing trajectories. The approach of Csanady [Csa63b, Csa63a] is essentially based on an extended form of Taylor’s hypothesis according to which the Eulerian temporal velocity correlation in the streamwise direction approaches the corresponding Eulerian spatial velocity correlation as the ratio of turbulent velocity fluctuations to mean velocity approaches zero. Within Csanady’s extension it is assumed that in a frame of reference moving with the mean flow, the La- grangian spatial velocity correlation can be approximated by the Eulerian temporal velocity correlation. Both theoretical and experimental support to this approximation exist. However, by analogy with Taylor’s hypothesis,

61 1. Equations of multiphase ﬂow mean relative velocity of particles exceeding the standard deviation of velocity ﬂuctuations at least by a factor of four is required for the approximation to be valid. In hes study of atmospheric dispersion Csanady used a coordinate system where x1 is the horizontal component (direction of atmospheric wind), x2 is the span-wise component and x3 is the vertical component (upwards). Based mainly on experimental observations [Csa63a] he expressed the Lagrangian spatial correlation functions in the three directions as

W τ Rλ = exp 3 (1.239) Lp,11 − L W τ W τ Rλ = 1 3 exp 3 (1.240) Lp,22 − 2L − L 2 λ τ 2 w3 RLp,33 = exp W + h i . (1.241) −Ls 3 β2    Here L is the characteristic size of an eddy (assumed to be approximately the same in all directions), W3 is the vertical component of mean relative velocity and w3 it’s ﬂuctuating part. The coeﬃcient β is given by

w2 T β = h 3i L , (1.242) p L where TL the Lagrangian time scale. Using the exponential approximation of the correlation function Eqn. (1.241) and Taylor’s theorem Eqn. (1.122) the mean square displacement in the vertical direction can be expressed in the non-dimensional form as 1 η = 2ξ 1 (1 exp( ξ)) , (1.243) − ξ − − where the dimensionless displacement η and the dimensionless time ξ are deﬁned by

y2 (t) w2 η = h L,3 i W 2 + h 3i w2 L2 3 β2 h 3i t w2 ξ = W 2 + h 3i . Ls 3 β2

The asymptotic long time vertical dispersion coeﬃcient now becomes (see Eqn. (1.128)) 2 w3 L lim Dp,3 = Dp∞,3 = h i . (1.244) t w2 →∞ 2 h 3 i W3 + β2 q 62 1. Equations of multiphase ﬂow

Therefore, at large mean relative velocity the mean square displacement and the long time dispersion coefficient are seen to be proportional to 1/W3. By dividing Eqn. (1.244) with the long time diffusivity of the fluid we finally obtain D∞ 1 p,3 = . (1.245) D 2 2 f∞,3 β W3 1 + w2 r h 3i Similarly, for dispersion in the horizontal and span-wise directions we find

2 w1 L Dp∞,1 = h i (1.246) W3 2 1 w2 L Dp∞,2 = h i , (1.247) 2 W3 and

D∞ 1 p,1 = (1.248) D 2 2 f∞,1 β W3 1 + w2 r h 3 i D∞ 1 p,2 = . (1.249) D 2 2 f∞,2 4β W3 1 + w2 r h 3i Notice, however, that the equation for horizontal direction is derived here exactly as for vertical direction, Eqn. (1.245), whereas the equation of lateral direction is written purely by analogy so as to give the correct behavior in the limits W 0 and W . 3 → 3 → ∞

63 2. Numerical methods

2.1 Introduction

Since the various multiphase flow models introduced in the previous chapter are, in general, complex set of heavily coupled nonlinear equations it is impossible, without dramatic simplifications, to get an analytic solution to any of those models. Without an exception numerical methods are needed. Most of the numerical work done on the multiphase flow equations has been focused to particle tracking algorithms (see section 1.5 for the theory of particle tracking) and solution of multifluid equations (1.57) and (1.58). We thus restrict ourselves to these traditional topics in this monoghraph. We also give a brief introduction to novel numerical methods that are especially suitable for direct simulation of certain types of multiphase flows in a ’mesoscopic’ scale. These include the lattice-gas, the lattice-Boltzmann and the dissipative particle dynamics methods. Until now, the numerical development of multifluid equations is done mainly within the finite difference method (FDM) or within the finite volume method (FVM). Applications using finite element method (FEM) seem to be less extensively studied. Independently of the applied discretization method the nature of the multifluid equations will end up to the same difficulties in the numerical solution procedures. While having the same mathematical form as the one phase Navier-Stokes equations also the same problems are encountered, including the problems of pressure-velocity coupling and dominating convection. Furthermore in multifluid flow equations, the inter-phase coupling terms pose specific demands to the numerical algorithms. Contrary to the multifluid flow models the particle tracking reveals no specific theoretical difficulties in numerical methods. The particle tracks are computed by numerical integration of the equation for particle motion and then the set of ordinary differential equations is solved along these tracks. It is a relatively easy method with some technical tricks and can be adopted to any existing one phase flow solver. As compared to conventional methods discussed above, mesoscopic simulation methods provide a completely different approach to multiphase flows as they do not resort to solving the averaged continuum equations. Instead, they are based on describing the fluid in terms of a large number of ’particles’

64 2. Numerical methods that move and collide in a discrete lattice. In sections 2.2 and 2.3 numerical solution algorithms based on the finite volume method (FVM) and finite element method (FEM) of the continuum multifluid flow equations are represented. In section 2.2 FVM algorithms and schemes are given in detail using Body-Fitted Coordinates. Also the methods, like pressure correction algorithm SIMPLE, mimicked from the one phase context are extended to cover multifluid equations. In section 2.3 the stabilized FEM algorithms are introduced. The presentation covers twofluid flow equations but can easily be extended to multifluid flow equations. In section 2.4 an overview of the basic numerical methods needed in particle tracking and finally in section 2.5 a brief introduction to mesoscopic numerical methods is given.

Balance equations In what follows we will consider the following continuum multifluid flow equations ∂ (φ ρ˜ ) + (φ ρ˜ u¯ ) = 0 (2.1) ∂t α α ∇ · α α α ∂ (φ ρ˜ u¯ ) + (φ ρ˜ u¯ u¯ ) φ µ [ u¯ + ( u¯ )t] ∂t α α α ∇ · α α α α − ∇ · α α ∇ α ∇ α Np = φ p˜+ B (u¯ u¯ ) + φ F˜ . − α∇ αβ β − α α α βX=1 These equations are derived from the general multifluid equations (1.57) and (1.58) neglecting the pseudo-turbulent stress and assuming that all the phases behave like Newtonian fluids, i.e., τ = φ µ [ u¯ + ( u¯ )t]. (2.2) ∇ · h αi α α ∇ α ∇ α Furthermore the momentum exchange term Dα is assumed to be due to inter-phase drag only, i.e.,

Np D = B (u¯ u¯ ). (2.3) α αβ β − α βX=1 Using a general dependent variable Φα the ﬂow equations for phase α can be given in a generic form ∂ (φ ρ˜ Φ ) + [φ (˜ρ u¯ Φ Γ Φ )] (2.4) ∂t α α α ∇ · α α α α − α∇ α Np = BI (Φ Φ ) + S . αβ β − α α βX=1 The continuity equation and the momentum equation for phase α follow from Eqn. (2.4) by setting Φα = 1 and Φα = u¯α, respectively.

65 2. Numerical methods

2.2 Multiﬂuid Finite Volume Method

The development of numerical methods for multifluid flow equations within FVM methods is based heavily on the extensions of single phase flow algorithms, such as the pressure-correction algorithm SIMPLE and its improve- ments to the multifluid context [Spa77, Spa80, Spa83, Kar02]. As discussed in the previous chapter, multifluid flow equations typically include for each phase equations that are formally similar to the conventional single fluid equations (unless simplified models such as the mixture model is considered), except of additional terms that arise due to the presence of other phases. These terms provide coupling between the equations and pose one of the major problems for modelling and for numerical solution of the multifluid flow equations. The coupling terms are often expressed in the form of a coupling constant times the difference between the values of the dependent variable of the two phases. When the coupling is strong the coupling constant is large and the value of the dependent variables in different phases are nearly equal. This condition with a large number multiplying a very small number often leads to convergence problems. The Partial Elimination Algorithm (PEA) developed by Spalding [Spa80] for twophase equations and the SINCE algorithm [Lo89, Lo90] for multifluid equations handles this problem by separating the solution of the phases of the dependent variable. This can be done effectively in the cell level. Further details of SINCE algorithm can be found in Ref. [KL99]. Until now the multifluid algorithms in FVM are based on structured grid approach. Complex geometries are handled by changing to Body Fit- ted Coordinates (BFC) and by using multi-block grids. A better way to approximate the physical geometry would be to use an unstructured grid where local refinements and adaptation are possible. Also, the ease of grid generation in complex geometries using automatic generation algorithms is a tempting feature of unstructured grids. Within FVM, the development of unstructured grid solvers has been distinct from the development of structured solvers where the benefits of the grid structure are exploited. More recently, development of unstructured grid FVM solvers has followed same guidelines as the development of structured grid solvers. In this method [MM97] the values are stored, like in structured solvers, in the cell centers and the use of the so called reconstruction gradient enables the construction of higher order schemes. Earlier, the problem of pressure oscillation was dealt with a staggered grid for velocity components and pressure. Recently this unwanted staggering (especially in complex geometries) is overcome using the Rhie-Chow algorithm [RC83, BW87], where the cell-boundary velocity components are obtained by interpolating the discrete momentum equations instead of the velocity values. In Control Volume Finite Element Method (CVFEM) by Baliga and Patankar [BP80] finite volume equations and element by element assembling, typical to FEM, are combined. In this

66 2. Numerical methods method the variables are attached to discretization points similarly to FEM. For this method some development is done also in twophase context [MB94]. In this Section the FVM method in general Body-Fitted Coordinates (BFC) for multifluid equations is represented. Pressure stabilization is dealt with the Rhie-Chow algorithm, where the cell-boundary velocity components are obtained by interpolating the discrete momentum equations instead of the velocity values. Three different treatments of the inter-phase coupling are given including explicit algorithm, PEA algorithm for two phase situation and an extension of the PEA algorithm to multifluid equations. The solution of the volume fraction equations is obtained from the scheme which ensures the volume fraction sum to unity. The pressure-velocity coupling is handled with the SIMPLE method which extension to multifluid equations is given in detail. In the end the overall algorithm called IPSA (Inter Phase Slip Algorithm) for the solution of the system of multifluid flow equations is given.

2.2.1 General coordinates In order to facilitate the numerical solution of the macroscopic balance equations (2.1), (2.3) and (2.4) in a general 3D geometry using Body-Fitted Co- ordinates (BFC), these equations are expressed in the covariant tensor form. The coordinate system used in this context is the local non-orthogonal coordinate system (ξ1, ξ2, ξ3), referred to as the computational space, obtained from the Cartesian coordinate system (x1, x2, x3), i.e. the physical space, by the curvilinear coordinate transformation xi(ξj) (Fig. 2.1). Because there is no danger of confusion, the phase and mass-weighted phase average notations above, e.g.,ρ ˜α and u¯α, have been omitted in the subsequent treatment in order to enhance the readability of the equations. Thus the balance equations in the computational space are ∂ ∂ ( J φ ρ ) + (Iî ) = 0 (2.5) ∂t | | α α ∂ξi mα i j m ∂ k ∂ î i ∂p ∂ AmAk ∂uα ( J φαραuα) + (I k ) = φαAk + (φαµα ) ∂t | | ∂ξi uα − ∂ξi ∂ξi J ∂ξj | | Np + J B (uk uk ) + J φ F k (2.6) | | αβ β − α | | α α βX=1 Np ∂ ∂ ( J φ ρ Φ ) + (Iî ) = J B (uk uk ) + J S , (2.7) ∂t | | α α α ∂ξi Φα | | αβ β − α | | α Xβ=1 where J is the Jacobian determinant of the mapping xi ξj(xi) and the | | î î î → total normal phasic fluxes I , I k and I are defined as mα uα Φα î i Imα = φαραuˆα, (2.8)

67 2. Numerical methods

ξ2

x ξ(x) →

2 x2 ξ ξ1

1 x ξ1

Figure 2.1: A single block of a collocated BFC-grid in the physical space (x1, x2, x3) and in the computational space (ξ1, ξ2, ξ3) with the dummy cells on the boundaries.

i j k î i k AmAm ∂uα I k = φα(ραuˆαuα µα ), (2.9) uα − J ∂ξj | | Ai Aj ∂Φ Iî = φ (ρ uî Φ µ m m α ). (2.10) Φα α α α α − α J ∂ξj | | i i m i The notationsu ˆα = Amuα and Am in the above equations stand for the normal flux velocity component and the adjugate Jacobian matrix of the mapping xi ξj(xi). The area vectors of the surfaces of an elementary → grid cell A(i) (Fig. 2.2), i.e., the vectors pointing to the outward normal direction of the cell faces and having the magnitude of the length equal to the magnitude of the area of these faces, are obtained as

A(1) = e e , A(2) = e e and A(3) = e e , (2.11) (2) × (3) (3) × (1) (1) × (2) where e(i) are the contravariant basis vectors associated with the curvilinear frame ξi(xj) (tangential to the coordinate curves). Covariant basis vectors (normal to the coordinate surfaces) are denoted by e(i). The cartesian components of contravariant vectors are given by

k e(i)k = Ji . (2.12)

With the help of these area vectors the volume of elementary grid cell VP is then related to the Jacobian determinant as

A(i) e = e e e δi = V δi = J δi . (2.13) · (j) (1) · (2) × (3) j P j | | j 68 2. Numerical methods

Because the two frames of basis vectors are dual to each other e(i) e = δi · (j) j the Eqn. (2.13) can be written in the form

A(i) e = J e(i) e . (2.14) · (j) | | · (j) (i) It can be shown that the cartesian components of the area vectors Ak are determined by the adjugate Jacobian matrix as

(i) i Ak = Ak. (2.15) The necessary information to perform the coordinate transformation includes only the volumes and the cartesian components of the area vectors of the grid cells in the physical space. This information is completed by calculation of the Jacobian determinant J and the adjugate Jacobian matrix i | | Ak of the transformation.

(2) A ξ2

n 1 W x ξ(x) → w D d u U (1) P A e

e 1 (2) E A(3) s e(1) ξ1 e(3)

1 ξ3 S

Figure 2.2: Area vectors on the faces of the ﬁnite-volume cell in the physical space and the notation of the neighboring cell centers and cell faces in the computational space.

2.2.2 Discretization of the balance equations Within the coordinate transformation all the derivatives of the macroscopic balance equations in the physical space have been expressed with the corresponding terms in the computational space, in which the transformed macroscopic balance equations are discretized. Now we are free to use any discretization method we prefer. In what follows the conservative ﬁnite volume approach with collocated dependent variables is applied.

69 2. Numerical methods

Integration of the generic balance equation over a single ﬁnite volume cell (Fig. 2.2) in the computational space and the use of Gauss’ law results in

Np ∂ ˆi I ( J φαραΦα) dVc + IΦα n dAc = J Bαβ(Φβ Φα) dVc Vc ∂t | | Ac · Vc | | − Z Z Z βX=1 + J S dV , (2.16) | | α c ZV c where Vc is the volume of cell and Ac is the surface of that cell in the computational space. In a same way the macroscopic balance equations (2.5), (2.6) and (2.7) can be formally represented as

t+∆t u n e J φαρα ˆ1 ˆ2 ˆ3 {| | } + Im + Im + Im = 0 (2.17) ∆t α d α s α w t t+∆t h i h i h i k u n e J φαραuα ˆ1 ˆ2 ˆ3 ∂p {| | } + I k + I k + I k = φα J uα uα uα i ∆t t d s w − | |∂ξ h i h i hu i n A1 Aj ∂um A2 Aj ∂um + φ µ m k α + φ µ m k α α α J ∂ξj α α J ∂ξj " | | #d " | | #s e A3 Aj ∂um + φ µ m k α + J φ F k α α J ∂ξj {| | α α } " | | #w Np + J B (uk uk ) (2.18) {| | αβ β − α } βX=1 t+∆t u n e J φαραΦα ˆ1 ˆ2 ˆ3 {| | } + IΦα + IΦα + IΦα ∆t t d s w h i h i h i Np = J BI (Φ Φ ) + J S . (2.19) {| | αβ β − α } {| | α} βX=1 In the above equations the notation indicates that the operand is in- {} tegrated over the volume of the finite volume cell in question. Because the computational space cells all are of unit volume, the resulting mean value u is directly expressed per unit volume basis. In addition, the notation [ ]d expresses a difference between the values of the operand at the specified faces or time, e.g., [Ii ]u = Ii Ii . (2.20) Φα d Φα u − Φα d When spatial integration over cell face is considered the right hand side notation Ii implies that the operand is integrated over the respective Φα u face (face corresponding to point u in this case) of the finite volume cell.

70 2. Numerical methods

Integration over finite volume cell It is seen from the Eqns. (2.17), (2.18) and (2.19) that the integration of the operand over the finite- volume cell noted as Φ is applied to the { } terms including time derivatives and to the source terms. To perform the integration the real field of the operand inside the cell is approximated with a constant value found at the center of the cell. So we apply a one point integration rule. Because the discretization is done in the computational space the integration results simply in

Φ Φ V = Φ = Φ . (2.21) { } ≡ |P c |P P Integration over cell surface The rest of the terms expressing the flux of the operand through the faces of the finite volume cell are integrated over the cell face. For the averaging the real profile of the operand over the face is approximated with a constant value located at the center of the face. Thus we apply again one point integration rule which results in

i Φ dAc Φ c Ac = Φ c = Φc. (2.22) i ≡ | | ZAc Now the integral forms of the phasic ﬂuxes can be represented as

[Iî ]h = Ci Ci (2.23) mα l α h − α l k k î h i k i k ij ∂uα ij ∂uα [I k ]l = Cα uα Cα uα Dα + Dα uα h h − l l − h ∂ξj l ∂ξj h l ∂Φ ∂ Φ î h i i ij α ij α [IΦα ]l = Cα Φα h Cα Φα l DΦ + DΦ , h | − l | − α h ∂ξj α l ∂ξj h l where 1 ; h = u, l = d i = 2 ; h = n, l = s (2.24)   3 ; h = e, l = w indicate the pair of faces in question. The phasic convection and diffusion coefficients Ci , Dij and Dij are specified by the relations α α Φα Ci = φ ρ uî (2.25) α c α|c α|c α c ij Dα = φα µα G ij (2.26) c |c |c |c ij DΦ = φα c Γα c Gij , (2.27) α c | | |c where 1 ; c = u, d i = 2 ; c = n, s (2.28)   3 ; c = e, w

 71 2. Numerical methods and Ai Aj G = m m . (2.29) ij J | | Values on cell faces u In order to calculate the ﬂux terms [Φ]d , the material properties and the values of the dependent variables at the centers of the cell faces are needed. These values are found by a weighted linear interpolation scheme from the cell center values. The weight factors used in this context are based on the distances between cell centers and corresponding cell faces on the physical space (Fig. 2.3). The value on the cell face is then given by

Φ = Φ = (1 W )Φ + W Φ , (2.30) |c c − c P c C where ∆P c W = (2.31) c ∆c P + ∆Cc and where c = (u, d, n, s, e, w) and C = (U,D,N,S,E,W ). This scheme is second order accurate in rectangular non-uniform meshes.

UN N DN n DN N UN x ξ(x) u U → P n d D s D d P u U U S S s DS DS S U S ∆uU u U Pu ∆ PU ∩

Figure 2.3: Illustration of the notations used in weighted interpolation.

Calculation of gradient at cell center The source terms include the pressure gradients calculated at the cell centers. These gradients are approximated by the same type of weighted interpolation scheme like the one for values on the cell faces, but in order to enhance

72 2. Numerical methods the accuracy the weight factors are in this case based on the arc lengths between the cell centers (Fig. 2.3). Although the calculation of the arc lengths is computationally expensive, the Rhie-Chow interpolation method which is i used to provide the normal ﬂux velocity componentsu ˆα on the cell faces requires second order accuracy also on the non-uniform curvilinear meshes. Thus the gradient at the cell center is obtained as ∂Φ = W Φ + (W W )Φ W Φ , (2.32) ∂xi L H H − L P − H L P

where the weight factors are deﬁned to be HP W = ∩ (2.33) H LP ( HP + PL) ∩ ∩ ∩ PL W = ∩ . (2.34) L PH ( HP + PL) ∩ ∩ ∩ Above the length of the arc for example from the point P to point L is denoted by PL. According to the Eqn. (2.32), the pressure gradient ∩ source term in the discretized momentum equation is not calculated on the computational space but directly from the physical space gradient.

Calculation of gradient on cell faces The total normal phasic fluxes include gradients of the dependent variable on the cell faces in their terms involved with diffusion fluxes. These gradients are all discretized using central differences on computational space. For the gradients normal to the cell faces the central difference method is applied as

1; C = U, c = u ∂Φ = Φ Φ with i = 2; C = N, c = n (2.35) ∂ξi C − P  c 3; C = E, c = e. 

In the case of cross-derivatives, i.e., derivatives on the plane of the cell face, the gradients are approximated as the mean of the two central diﬀerences calculated at the cell centers on the both sides of the face in question. As an example, the cross-derivatives on the cell face u are (Fig. 2.3)

∂Φ = 1 (Φ Φ + Φ Φ ) (2.36) ∂ξ2 4 N − S UN − US u ∂Φ = 1 (Φ Φ + Φ Φ ). (2.37) ∂ξ3 4 E − W UE − UW u

Approximation of the cross-derivatives connects eighteen neighboring cells to the treatment of every ﬁnite volume cell. To reduce the bandwidth of the coeﬃcient matrix the treatment is reduced back to the usual one involving only the eight neighboring cells sharing a cell face with the cell under

73 2. Numerical methods consideration by the deferred correction approach, i.e., treating the cross- derivatives as source terms by using values from the previous iteration for the dependent variables in question. Thus in addition to the terms related to the gradients normal to the cell faces (the diagonal terms of the diffusion tensor), the deferred correction approach results to the following additional source terms expressing the effect of non-orthogonality ∂?Φ ∂?Φ u ∂?Φ ∂?Φ n SD = D12 α + D13 α + D21 α + D23 α Φα Φα ∂ξ2 Φα ∂ξ3 Φα ∂ξ1 Φα ∂ξ3 d s ∂?Φ ∂?Φ e + D31 α + D32 α ,(2.38) Φα ∂ξ1 Φα ∂ξ2 w ? where Φα denotes the value of the dependent variable from the previous iteration. Following the details given above the formally discretized macroscopic balance equations (2.17), (2.18) and (2.19) can now be written in the final form. To enhance the readability and to keep more physical nature in the discretized equations they are next given in a partly discretized form retain- ing most of the terms in their previous form. Thus the macroscopic balance equations of mass, momentum and generic dependent scalar variables are φ ρ V t+∆t C1 C1 + C2 C2 + C3 C3 = α|P α|P P (2.39) α u − α d α n − α s α e − α w − ∆t t

k k ∂p auk uα = auk uα φα P VP α c P α c C − | ∂xk c c,C P X X u n e 1 j 2 j 3 j A A ∂um A A ∂um A A ∂um + φ µ m k α + φ µ m k α + φ µ m k α α α J ∂ξj α α J ∂ξj α α J ∂ξj " | | #d " | | #s " | | #w Np 0 k k k + φα P Fα VP + Bαβ (uβ uα )VP | P |P P − P β=1 X k D φα P ρα P uα P VP +Suk | | (2.40) α − " ∆t # t Np a Φ = a Φ + BI (Φ Φ )V Φα |c α|P Φα |c α|C αβ P β|P − α|P P c c,C β=1 X X X 0 + Sα P VP φ ρ Φ V +SD α|P α|P α|P P . (2.41) Φα − ∆t t 0 k 0 In the Eqn. (2.41) Fα P and Sα P are the constant part of the linearized source terms. The summation symbols are deﬁned as

Φ = Φ + Φ + Φ + Φ + Φ + Φ (2.42) |c |u |d |n |s |e |w c X 74 2. Numerical methods

a Φ = a Φ + a Φ + a Φ |c |C |u |U |d |D |n |N Xc,C + a Φ + a Φ + a Φ . (2.43) |s |S |e |E |w |W With hybrid differencing scheme the matrix coefficient from convective and diffusive transport above can be written as

1; h = u a = max 1 Ci , Dii 1 Ci i = 2; h = n Φα |h 2 α h Φα h − 2 α h  3; h = e  (2.44) 1; l = d a = max 1 Ci , Dii + 1 Ci i =  2; l = s Φα |l 2 α l Φα l 2 α l  3; l = w 

In the hybrid differencing scheme the central differencing is utilized when the ii cell Peclet number (Peα = Cα/DΦα ) is below two and the upwind differencing, ignoring diffusion, is performed when the Peclet number is greater than two. This scheme together with the deferred correction approach associated with the gradients on the cell faces guarantees the diagonal dominance of the resulting coefficient matrix. For that reason hybrid differencing scheme serves as the base method for which other more accurate schemes can be built upon by the deferred correction approach [Com, LL94].

2.2.3 Rhie-Chow algorithm In calculating the convection ﬂuxes across cell faces, velocities have to be inferred from those calculated at cell centers. A straightforward linear interpolation, i.e.,

ui = u i = (1 W ) ui + W ui (2.45) α e α|e − e α P e α E would lead to the well known chequer-board oscillations in the pressure field, since the use of 2δξ-centered differences for the computation of pressure gradients at control cell centers effectively decouples the even and odd grids. This problem is overcome by the Rhie-Chow algorithm where the cell boundary velocities are obtained by interpolating the discrete momentum equations. k k The Cartesian velocity components uα P , uα E at control volume centered at P and at E correspondingly obey the discretized momentum equation (2.40) given in short hand form as

i k ∂p i uα + bαi = Sα (2.46) P P ∂xk P P

i k ∂p i uα + bαi = Sα , (2.47) E E ∂xk E E

75 2. Numerical methods where k k φα C Ai bαi = | (2.48) C aui c α c j u j n j e A1 A ∂um A2 A ∂um A3 A ∂um i P m i α m i α m i α Sα C = φαµα j + φαµα j + φαµα j " J ∂ξ # " J ∂ξ # " J ∂ξ # | | d | | s | | w Np 0 i i i D + φα Fα VP + Bαβ (u uα )VP + S i |P P |P β P − P uα β=1 X i t+∆t φα P ρα P uα P VP | | a i . (2.49) − ∆t  uα c " # c ! t . X In a staggered grid approach the pressure oscillation is avoided by discretiz- ing the momentum equation on the grid whose centers are the faces of the original grid. In this case the face velocity component obeys the discretized momentum equation

i k ∂p i uα + bαi = Sα , (2.50) e e ∂xk e e where the pressure gradients are calculated using 1δξ-centered diﬀerences. In the Rhie-Chow method the solution of Eqn. (2.50) is approximated interpolating linearly momentum equations (2.46) and (2.47). Approximat- ing the right side term in Eqn. (2.50) by weighted linear interpolation of the corresponding terms in Eqns. (2.46) and (2.47) we get

i k ∂p i i k ∂p uα + bαi = Sα e = uα e + bαi . (2.51) e e ∂xk | | e ∂xk e e

Assuming that b k b k , we obtain αi e ≈ αi e ∂p ∂p u i = ui + b k . (2.52) α e α|e αi e ∂xk − ∂xk e e!

i i k For the normal velocity componentsu ˆα = Ak uα we get

i i ˆik ∂p ∂p uˆα = uˆα e + bα , (2.53) e | e ∂xk − ∂xk e e! where

ˆik i k bα = Aj bαj . (2.54) e e Since all pressure gradients are computed using central diﬀerences, the cross- derivative terms in Eqn. (2.53) cancel leaving the formula

i i ˆii ∂p ∂p uˆα = uˆα e + bα . (2.55) e | e ∂xi − ∂xi e e!

76 2. Numerical methods

2.2.4 Inter-phase coupling algorithms The macroscopic phasic balance equations (2.39), (2.40) and (2.41) are coupled to the corresponding balance equations of the other phases with the transport related to the change of phase and with the interfacial force density. In many cases these couplings are strong which results into a very slow convergence when iterative sequential solution methods are used. To solve this problem some degree of implicitness is required in the treatment of the interfacial coupling terms. The approaches are introduced here using the equation of the generic dependent variable (2.41).

Explicit treatment The only necessary operation in this approach is to transfer all the terms depending on the variable to be solved, Φα, to the left side of Eqn. (2.41) (implicit handling of Φα) and leave the other terms on the right side (explicit handling of Φβ). The resulting linear equation can be expressed in the following compact form Φ = (2.56) AΦα α|P CΦα Np = a + BI V 1S V AΦα Φα |c αβ P P − α P P c β=1 X X φ ρ V + α|P α|P P (2.57) ∆t Np = a Φ + BI Φ V CΦα Φα |c α|C αβ P β|P P c,C β=1 X X φ ρ Φ V + 0S V + SD + α|P α|P α|P P . (2.58) α P P Φα ∆t t

Partial Elimination Algorithm (PEA) When there are only two phases to be considered simultaneously, the phasic balance equations of the dependent variables Φα and Φβ can be written in the form where they can be mathematically eliminated from the balance equations of each other. The equations for the phases α and β are first written as Φ = I (Φ Φ ) + (2.59) AΦα α|P Bαβ β|P − α|P CΦα Φ = I (Φ Φ ) + , (2.60) AΦβ β|P Bαβ α|P − β|P CΦβ where the coefficient , , , and are defined by the equa- AΦα AΦβ Bαβ CΦα CΦβ tions φ ρ V = a 1S V + α|P α|P P (2.61) AΦα Φα |c − α P P ∆t c X

77 2. Numerical methods

φβ ρβ VP = a 1S V + |P |P (2.62) AΦβ Φβ c − β P P ∆t c X I = BI V (2.63) Bαβ αβ P P = a Φ + 0S V CΦα Φα |c α|C α P P c,C X φ ρ Φ V +SD + α|P α|P α|P P (2.64) Φα ∆t t

= a Φ + 0S V CΦβ Φβ c β|C β P P c,C X φ β ρβ Φβ VP +SD + |P |P |P . (2.65) Φβ ∆t t

Eqns. (2.59) and (2.60) can be rearranged to the form

( + I )Φ = I Φ + (2.66) AΦα Bαβ α|P Bαβ β|P CΦα ( + I )Φ = I Φ + . (2.67) AΦβ Bαβ β|P Bαβ α|P CΦβ

Now it is clearly seen that when the inter-phase transfer coeﬃcient I is Bαβ very large, the values of the the two phases are very close to each other. Furthermore, in the sequential iterative solution the variables would have changed very little from their starting values and thereby the rate of convergence is slow. PEA algorithm manipulates the above equations to eliminate Φα and Φβ from the Eqns. (2.59) and (2.60) giving

I + Bαβ ( + ) Φ AΦα AΦα AΦβ α|P AΦβ ! I = Bαβ ( + ) + (2.68) CΦα CΦβ CΦα AΦβ I + Bαβ ( + ) Φ AΦβ AΦα AΦβ β|P AΦα ! I = Bαβ ( + ) + . (2.69) CΦα CΦβ CΦβ AΦα Clearly the Eqns. (2.68) and (2.69) are now fully decoupled from each other and the problem of slow convergence related to the strong coupling of phases is eliminated. Though it should be noticed that in this approach only the interfacial force density can be incorporated implicitly in the algorithm but the transfer related to the change of phase would be treated as in the explicit method above.

78 2. Numerical methods

Simultaneous solution of Non-linearly Coupled Equations (SINCE) The PEA algorithm, developed for twophase flows above, can not be generalized for multifluid conditions. A straightforward method to include some implicitness in the treatment of the interfacial coupling terms also in multifluid flows has been developed by Lo [Lo89]. The generic macroscopic balance law for Np phases can be arranged like Eqns. (2.66) and (2.67) in the twophase case to the following system of linear equations

E Φ ? = C , (2.72) Φ |P Φ where Φ ? is the solution vector, E is the coeﬃcient matrix and C is the |P Φ Φ right hand side vector. Solving this system cell by cell the new estimates are obtained. These are next substituted for the interfacial coupling terms on the right side of Eqns. (2.70) resulting to the following explicit balance equation NP Φ = + I Φ ? . (2.73) DΦα α|P CΦα Bαβ β|P βX=1 2.2.5 Solution of volume fraction equations In the simpliﬁed form the continuity equations (2.39) can be written for the solution of volume fractions as

79 2. Numerical methods

The coefficient a , b above are defined as φα |c φα |c 1 ; h = u a = 1 ρ uî uî , i = 2 ; h = n (2.77) φα |h 2 α|h | α h | − α h  3 ; h = e  1 ; l = d  a = 1 ρ uî + uî , i = 2 ; l = s (2.78) φα |l 2 α|l | α l | α l  3 ; l = w 

To ensure that the volume fractions always sum to unity, i.e.,

NP φ = 1 (2.79) α|P αX=1 we can solve instead of Eqn.(2.74) the equation

Nα φα = . (2.80) P N Nβ φβDβ | D (1+ P − ) α β=1 Dβ P This equation is equivalent to the original equation when the constraint equation (2.79) is used. Furthermore, we can see that Eqn. (2.80) reduces to the form of Eqn. (2.74) when the residuals N φ D of all the phases α − α α α vanish.

2.2.6 Pressure-velocity coupling

k Because of the iterative nature of the solution procedure, the velocities uα obtained from the solution of the phasic momentum equations do not usually fulﬁll the phasic (2.39) and total mass balances when substituted into the corresponding continuity equations. Therefore a correction to the pres- k? sure δp is desired which produces new estimates of the phasic velocities uα obeying the total mass balance. These new estimates are denoted as a sum of the current value and a correction

k? k k uα = uα + δuα (2.81) k? k k uˆα =u ˆα + δuˆα (2.82) p? = p + δp. (2.83)

To limit the interdependency of the cells the velocity corrections to the neighbors of the cell under consideration are discarded. Accordingly, these velocities are approximated by

k? k k? k uα = uα + Wpc uα uα , (2.84) C C P − P

80 2. Numerical methods

where the weight factor Wpc = 0 for the SIMPLE algorithm and Wpc = 1 for the SIMPLEC algorithm [VDR84]. Substituting the Eqns. (2.81), (2.82), (2.83) and (2.84) to the momentum equation (2.40) results in

k? k Auk Wpc auk uα = auk uα α P − α c! P α c C c c,C X X i ∂p R k φα P Ak + s k Wpc auk uα − | P ∂ξi uα − α c P P c X ∂δp φ Ai , (2.85) − α|P k P ∂ξi P

where

Using the original equation (2.40) some terms are canceled from the Eqn. (2.86) and the desired corrections to the phasic Cartesian velocity components can now be written in a compact form

k i ∂δp δuα = αk , (2.88) P − H P ∂ξi P

where φ Ai i α|P k P k αk P = hα , (2.89) H VP P

and k VP hα = . (2.90) P Auk Wpc c auk α P − α c

The corresponding corrections to the phasic normalP flux velocity components are according to the definition of phasic normal flux

i ˆij ∂δp δuˆα = α , (2.91) P H P ∂ξi P

81 2. Numerical methods where 3 i j φα P Ak Ak îj i j | P P k α = Ak P αk = hα . (2.92) H P H P VP P k=1 X The corrections to the phasic normal flux velocity components produce also new estimates for the phasic convection coefficients (2.25), which can be expressed as

i ? i i i i Cα = Cα + δCα = φα c ρα c uˆα + φα c ρα c δuˆα . (2.93) c c c | | c | | c

Finally to get the desired equation for the pressure correction the total mass balance is formed by adding all the phasic mass balances (2.39) together. Utilizing Eqn. (2.93) the joint continuity is given by

NP δC1 δC1 + δC1 δC1 α u − α d α n − α s α=1 X h + δC 1 δC 1 + / ρ = 0, (2.94) α e − α w Rmα α|P i where mα denotes the phasic mass residual with the current convection R i coeﬃcients Cα

φ ρ V t+∆t = α|P α|P P +C1 C1 + C2 C2 + C3 Rmα ∆t α u − α d α n − α s α e t C3 . (2.95) − α w

As shown by Eqn. (2.94), the mass balance of each phase is normalized by its phasic density before the summation in order to avoid bias of the correction procedure towards the heavier ﬂuid. Evidently the total mass balance (2.94) can now be written for the corrections of the phasic normal ﬂux velocity components as

NP φ ρ δuˆ1 φ ρ δuˆ1 + φ ρ δuˆ2 α|u α|u α u − α|d α|d α d α|n α|n α n α=1 X h φ ρ δuˆ2 + φ ρ δuˆ3 φ ρ δuˆ3 − α|s α|s α s α|e α|e α e − α|w α|w α w + ) / ρ = 0. (2.96) Rmα α|P i In Eqn. (2.96) the corrections of the phasic normal flux velocity components are introduced on the cell faces. This necessitates interpolation of these values from the cell centers where they are originally determined (see Eqn. i (2.91). The adjugate Jacobian matrix Ak is naturally defined on the cell face k but the coefficient hα has to be interpolated following the weighted linear interpolation procedure introduced by Eqn. (2.30). If the pressure correction

82 2. Numerical methods gradients on the cell faces are approximated using central differences on the computational space (2.35)–(2.37), Eqn. (2.91) can be substituted in the joint continuity equation (2.96) to give the desired pressure correction, namely NP A δp = a δp + SD Rmα , (2.97) p|P |P p|c |C p − ρ α=1 α P Xc,C X | where A = a (2.98) p|P p|c c X N îj P φα c ρα c α a = | | H c . (2.99) p|c ρ α=1 α P X | D The term Sp in Eqn.(2.97) denotes the cross-derivatives of the shared pressure corrections treated according to the deferred correction approach. Using the formal discretization notation it can be written as ∂?δp ∂?δp u ∂?δp ∂?δp n SD = E12 + E13 + E21 + E23 (2.100) p p ∂ξ2 p ∂ξ3 p ∂ξ1 p ∂ξ3 d s ∂?δp ∂?δp e + E31 + E32 , p ∂ξ1 p ∂ξ2 w where the pressure correction coefficients are defined as

N ˆij P φα c ρα c α Eij = | | H c . (2.101) p ρ α=1 α P X | 2.2.7 Solution algorithm The discretized macroscopic balance equations of mass, momentum and generic scalar dependent variable (2.39), (2.40) and (2.41) are solved itera- tively in a sequential manner. The solution methods studied are extensions of the well known single phase solution algorithm SIMPLE (Semi-Implicit Method for Pressure-Linked Equations) of Patankar & Spalding [PS72].

Inter Phase Slip Algorithm (IPSA) The IPSA method [Spa77, Spa80, Spa83] is based on the following sequential solution steps repeated until the convergence is achieved: Algorithm 1

1. Solve momentum equations for all the phases using values for the dependent variables obtained from the initial guess or from the previous iteration cycle.

83 2. Numerical methods

2. Solve the pressure correction equation (2.97) based on the joint continuity equation (2.94).

3. Update the pressure (2.83).

4. Correct the phasic Cartesian velocity components (2.88).

5. Correct the normal ﬂux velocities at cell centers (2.91).

6. Calculate the normal ﬂux velocity components at cell faces with the Rhie-Chow interpolation method (2.55).

7. Calculate the phasic convection (2.25) coeﬃcients.

8. Solve the phasic continuity equations for the phasic volume fractions from Eqn.(2.80).

9. Solve the conservation equations for the other scalar dependent variables.

As shown by Eqn. (2.85), in the IPSA method the interphase coupling terms are treated as in the explicit algorithm (2.56)– (2.58). In the IPSA-C method [KL99] the performance of the pressure correction step has been improved by incorporating the interphase coupling terms into this step following the idea of the SINCE method. In cases where the coupling of the phases is strong this improves convergence.

2.3 Multiﬂuid Finite Element Method

In finite element methods the development of multifluid flow algorithms must also be based on the one phase algorithms. For one phase flow it is well known that the standard Galerkin methods may suffer from spurious oscillations when applied directly to Navier-Stokes equations. This is due to two main reasons. The first reason is the advection-diffusion character of the equations, when oscillations contaminate the dominating advection. The second reason is the mixed formulation character which, with unsuit- able choice of pressure-velocity interpolations, may lead to pressure oscillations, similarly to FVM. These shortcomings can be dealt with proper choice of interpolation pair fulfilling the ’inf-sup’ condition and modifying the advective operator to include some ’upwinding’ effect. Recently, great interest have been taken in the so-called stabilized finite element methods [FFH92, FF92, FHS92], where these two problems can be handled by adding extra stabilizing terms into the system of variational equations. The consistency is preserved within these methods since the stabilizing terms will vanish with the residual of the governing equations. This stabilizing algorithm can be extended to cover twofluid flows [Hil97]. When applied a

84 2. Numerical methods strongly coupled system arises. Besides the inter-phase coupling we have an extra coupling, although not so tight, appearing through the stabilizing terms. The use of PEA like algorithms reducing the inter-phase coupling is not possible because of the nonlocal character of elementwise equations. To be able to solve this system properly all equations must be solved simultaneously. This amounts to need of more memory and computation time for a single iteration step but on the other hand when Newton based linearisation is used gives an optimal convergence rate and therefore less iteration steps are needed. In this Section we will represent a stabilized finite element algorithm for the solution of isothermal steady-state twofluid continuum equations. Let Ω RN, N = 2, 3 be the flow domain and let the carrier fluid be denoted ∈ by subscript f and the dispersed phase be denoted by subscript A. Let the phases be intrinsically incompressible, i.e.,ρ Ã = constant, ρ˜f = constant. Assuming stationary flow we get from Eqns. (2.1) and (2.3) (φ u¯ ) = 0 (2.102) ∇ · A A (φ u¯ ) = 0 ∇ · f f 2µ (φ Π(u¯ )) +ρ ˜ φ (u¯ )u¯ = φ p˜ + B(u¯ u¯ ) − A∇ · A A A A A · ∇ A − A∇ A − f + φAF˜ A 2µ (φ Π(u¯ )) +ρ ˜ φ (u¯ )u¯ = φ p˜ + B(u¯ u¯ ) − f ∇ · f f f f f · ∇ f − f ∇ f − A + φf F˜ f , where Π(u¯) = 1 ( u¯ +( u¯)t) is the rate of deformation tensor. For simplic- 2 ∇ ∇ ity, viscosities µα are assumed to be constant. For the inter-phase momentum transfer coefficient we have B = BAf = BfA. Next we will write Eqns. (2.102) in dimensionless form. To this end we introduce the dimensionless coordinates and independent variables. Let U be some characteristic velocity and L some characteristic length of the system. We define x? = x/L to be ? the dimensionless coordinate, uα = u¯α/U to be the dimensionless velocity ? 2 and p =p/ ˜ (˜ρf U ) to be the dimensionless pressure. With these notations Eqns. (2.102) can be written in a dimensionless form as ? ? (φAuA) = 0 (2.103) ∇ ?· ? (φf uf ) = 0 ? ? ∇? · ? ? ? ? ? ? ? (2/ReA) (φAΠ(uA)) + φA(uA )uA = β[ φA p + B (uf uA) − ∇ · · ∇ − ?∇ − +φAFA] ? ? ? ? ? ? ? ? ? ? (2/Ref ) (φf Π(uf )) + φf (uf )uf = φf p B (uf uA) − ∇ · · ∇ − ∇? − − +φf Fγ, where Reα =ρ ˜αUL/µα is the Reynolds number for phase α, β =ρ ˜f /ρÃ is ? ? the density ratio, is the dimensionless gradient operator, B = L/(¯ρf U)B ∇ ? 2 ˜ is the dimensionless momentum transfer coefficient and Fα = L/(U ρ˜f )Fα is the dimensionless body force.

85 2. Numerical methods

2.3.1 Stabilized Finite Element Method The traditional sequential methods, which are frequently used with the finite volume methods suffer from poor convergence when the inter-phase coupling is significant enough. In order to attain better convergence in such cases fully coupled equations must be solved. Although this will lead to a large algebraic system of equations and therefore will demand more memory and computation time, the achieved gain in convergence, especially when Newton-type linearization is used, will yield lower total cost in computing time. This is even more plausible in cases where efficient iterative solvers and especially matrix free algorithms are used. About the algorithmic procedures for effective solution strategies see the paper by Hughes and Jansen [HJ95]. Another advantage of the full coupling is the increased stability of the method. For Eqns. (2.103) stabilization must be applied at two levels, namely the stabilization of velocity pressure pair and the stabilization of phase volume fractions. In what follows a stabilized finite element method for solving the system of Eqns. (2.103) is introduced [Hil97]. Within the method the above mentioned problems of coupling and stabilization of the equations are resolved. To understand the following presentation of the method the reader needs some basic knowledge of the functional spaces. One can study these things from the standard FEM text book, for example [KN90]. In the sequel L2(Ω) 2 is the space of square integrable functions in Ω and L0(Ω) the space of functions in L2(Ω) with zero mean value in Ω. Furthermore ( , ) stands for the · · L2-inner product in Ω, e.g., for functions u, v V the L2-inner product is ∈ h defined as N (u, v) = uividx. (2.104) Ω Xi=1 Z Also, 0(Ω) is the space of continuous functions in Ω and H1(Ω) is the C 0 Sobolev space of functions with square integrable value and derivatives in Ω with zero value on the boundary of Ω. In order to formulate the stabilized finite element method we first specify the required functional spaces as follows

V = v H1(Ω)N : v P (K)N, K Π , h { ∈ 0 |K ∈ k ∈ h} P = p 0(Ω) L2(Ω) : p P (K), K Π , h { ∈ C ∩ 0 |K ∈ l ∈ h} where N is the dimension of the problem and k, l 1 are integers. The ≥ symbol Πh denotes the partition of Ω¯ into elements and h is the size of that partition. The symbol Pk(K) denotes the space of polynomial functions of degree k or less deﬁned on element K. Furthermore for volume fractions we introduce the space

Φ = φ H1(Ω) : φ P (K), K Π , h { ∈ |K ∈ m ∈ h} 86 2. Numerical methods where the integer m 1. In what follows, we drop the superscript ? in ≥ Eqns. (2.103). The stabilized method considered for Eqns. (2.103) can now be stated formally as: Method 2.3.1 Find (u , u , φ , p ) V V Φ P such that Ah fh Ah h ∈ h × h × h × h

A(uAh , ufh , φAh , ph; vA, vf , qf , qA) = F (vA, vf , qA, qf ), (2.105) (v , v , q , q ) V V Φ P , A f A f ∈ h × h × h × h with

A(u , u , φ , p; v , v , q , q ) = C (u ; v ) + D (u ; v ) A f A A f A f { α α α α α α αX=A,f convection viscous stress P (v ; p) P (u ; q ) I (u u ; v ) − α |α {z− α } α | α −{z α }β − α α pressure continuity interaction + S (u , p, u u ; v , q ) | α {zα } β|− {zα α } α | {z } momentum stabilization + SM (u , φ ; u , q ) + D¯ (φ ; q ) | α α α {zα α } } A A A } continuity stabilization discont. capturing

F (vA, vf , qA, qf ) = | Bα(v{zα, qα) } | {z } αX=A,f where

C (u; v) = (φ u u, v) α α · ∇ Dα(u; v) = ((2/Reα)φαΠ(u), Π(v)) D¯ (φ; q) = (κ φ, q) A ∇ ∇ P (v; q) = ( (β φ v), q) α ∇ · α α Iα(w; v) = (Bβαw, v) S (u, p, w; v, q) = (Res (u, p, w), τ (φ u v φ β q)) α α α α α · ∇ − α α∇ Res (u, p, w) = (2/Re ) (φ Π(u)) + φ u u + φ β p β Bw α − α ∇ · α α · ∇ α α∇ − α SM (u, φ; v, q) = (ResM (φ, u), δ (β qv)) α α α∇ · α ResM (φ, u) = (β φu) α ∇ · α G (v, q) = (φ β F˜ , v + τ (φ u v φ β q)) α α α α α α α · ∇ − α α∇ βα =ρ ˜f /ρ˜α.

The coeﬃcient κ for discontinuity capturing operator and the stability parameters τα, δα are deﬁned as

κ = c h Res /( U + h ) dif K TOT |∇ |p K Res = Res + Res + ResM + ResM TOT | A|1 | f |1 | A | | f | 87 2. Numerical methods

N U = ( u + u ) + φ + p |∇ |p |∇ Ai |p |∇ fi |p |∇ A|p |∇ |p Xi=1 δ = λ u h ξ(ReK(x)) f f | f |p K f δ = λ φ h ξ(PeK(x)) A A| A|p K A τ = hK ξ(ReK(x)) α 2 uα p α | | ReK(x) = 1 m u h Re α 4 K| α|p K α PeK(x) = 1 m u h /κ A 2 K| A|p K γ, 0 γ < 1 ξ(γ) = ≤ 1, γ 1 ≥ 1/p N p uα , 1 p < uα p = i=1 | i | ≤ ∞ | | ( max u , p = , P i=1,N | αi | ∞ where λα, cdif and mK are positive constants. Notice that the variational system (2.105) is nonlinear due to the convection and interaction terms and due to the discontinuity capturing terms and the stabilization terms. The stabilization terms Sα for momentum equations are similar to those used for incompressible Navier-Stokes equations (see, e.g., [FF92]). The only difference is that here the volume fractions are involved. The contribution arising from the lower order terms in stabilization, i.e., the interaction terms in the momentum equations, is neglected. The inclusion of interaction terms would lead to different stability parameters. Such a case is studied for diffusion-convection type scalar equation for example in ref. [LFF95]. Since we shall restrict ourselves to linear approximations here, the terms including second order derivatives are neglected. M The stabilization terms Sα arising from the continuity equations are different for each phase. For the carrier phase it affects directly the momentum equation while for dispersed phase it affects the continuity equation. That is also reason for the different normalization of the stabilization parameters δα. Since steep gradients can be expected in many applications of the method, we have introduced the discontinuity capturing operator D¯(φA, qA) to the equations. In the above method this term is added only for continuity equation of the dispersed phase. The corresponding diffusivity coefficient κ depends on the discrete residual of the system. The form of the coefficient equals to the one introduced by Hansbo and Johnson [HJ91]. The value of the coefficient is largest near steep gradients and vanishes when the solution is smooth. Stability analysis can be carried out for the linearized form of Eqns. (2.105) following the same guidelines as for incompressible Navier-Stokes equations (see, e.g., [FF92]), except of additional complications which arise from the interaction terms and from the compressible nature of the flow.

88 2. Numerical methods

The analysis is nevertheless nontrivial and is skipped here. In Galerkin Finite Element Method the solution functions

(uAh , ufh , φAh , ph) are approximated as a linear combination of basis functions (v˜ , v˜ , q˜ , q˜ ) in a ﬁnite-dimensional product space V V Φ P . A f A f h × h × h × h In other words, the phase velocities uk V , V = (V )N (componentwise) αh ∈ h h h are approximated as

dim Vh k k uαh(x) = uαjv˜j(x), k = 1,...,N (2.106) Xj=1 where uk R are the unknowns in discretization points x , j = 1,..., dim V . αj ∈ j h In a same way the dispersed phase volume fraction φAh and pressure ph are approximated as

dim Φh

φAh (x) = φAjq˜Aj(x) (2.107) Xj=1 dim Ph

ph(x) = pjq˜f j(x), (2.108) Xj=1 where φAj and pj are the nodal values of the dispersed phase volume fraction and pressure respectively. In the following we will assume that the unknowns are approximated using equal order elements and therefore the nodal values are collocated and if boundary effects (elimination of degrees of freedoms) are neglected the dimensions of the spaces equals, i.e., we can denote dim Vh = dim Φ = dim P n . h ≡ dofs In Galerkin approximation, the test functions (vA, vf , qA, qf ) and basis functions (vÃ, v˜f , qÃ, q˜f ) are taken from the same space. Let us then denote ϕ v˜ =q ˜ =q ˜ . Using definitions (2.106)–(2.108) the variational j ≡ j Aj f j system of equations (2.105) can be written in semi-discretized matrix form

AU = F, (2.109) where A is an (2 (N + 1) ndofs) (2 (N + 1) ndofs) matrix and F = × × t × × × (F1(k),F2(k),...,F2 (N+1)(k)) is the right hand side vector, where Fi(k) × 1 N 1 are ndofs dimensional subvectors. The vector U = (uf ,..., uf , p, uA,... N t , uA, φA) is the nodal unknown, where the subvectors are deﬁned by ui = (ui , . . . , ui ) (2.110) α α1 αndofs

p = (p1, p2, . . . , pndofs ) (2.111)

φA = (φA1, φA2, . . . , φAndofs ). (2.112) Note that the above system of equations is nonlinear in the sense that the coeﬃcient matrix A and the right hand side vector F depend on the solution U.

89 2. Numerical methods

2.3.2 Integration and isoparametric mapping The construction of the coefficient matrix A and right hand side vector F requires the computation of integrals over interpolation functions ϕj. In- stead of the global integration and global equations and global interpolation functions we can integrate in a single element with local equations and local interpolation functions. These local equations can then be assembled to a global system using special assembling algorithm. Details concerning these standard finite element procedures one can find in any textbook concerning finite element methods. One further standard procedure in finite element methods is the use of isoparametric mapping converting the physical coordinates into computational coordinates (compare with BFC in FVM). In finite element method the relationship between the physical coordinates (x1, x2, x3) and the computational coordinates (ξ1, ξ2, ξ3) for any element K in the partitioning Πh is obtained using parametric concept, i.e., a coordinate transformation defined by

K nnodes i j j i x (ξ ) = λk(ξ )xk, (2.113) Xk=1 i where λk are the interpolation functions over the element K and xk the nodal K coordinate values of that element. The symbol nnodes denotes the number of the nodes in element K. The above defined transformation is quite general. When the interpolation functions defining the dependent variables are of the same order as the functions defining the element geometry, i.e., ϕi = λi, the element is called isoparametric. If the order of λi is greater or less than that of ϕi the element is superparametric or subparametric correspondingly. For example the four node quadrilateral element in 2 dimension is isoparametric and we get

4 1 1 2 1 2 1 x (ξ , ξ ) = ϕ˜k(ξ , ξ )xk Xk=1 4 2 1 2 1 2 2 x (ξ , ξ ) = ϕ˜k(ξ , ξ )xk Xk=1 1 2 Note that the interpolation functionsϕ ˜k(ξ , ξ ) above are local functions in reference element, i.e., they are the same for every physical element (see Fig. 2.4). The derivatives are transferred according to the chain rule. In a matrix form we get ∂ϕ˜i ∂ϕi ∂ξ1 ∂x1 ∂ϕ˜i ∂ϕi  ∂ξ2  = J  ∂x2  , (2.114)  ∂ϕ˜i   ∂ϕi   ∂ξ3   ∂x3      90 2. Numerical methods where J is the Jacobian matrix of the mapping ξj xi(ξj). From this we → get ∂ϕi ∂ϕ˜i ∂x1 ∂ξ1 ∂ϕi 1 ∂ϕ˜i  ∂x2  = J−  ∂ξ2  , (2.115)  ∂ϕi   ∂ϕ˜i   ∂x3   ∂ξ3  1     where J− is the inverse of the Jacobian. When isoparametric elements are used, integration can be done over the reference element. Let us denote the reference element by E. Now we have for example for the local convection coeﬃcient C(K)(k, l) = (φ u ϕ , ϕ ) α α α · ∇ l k K = φ (x)u (x) ϕ (x)ϕ (x)dx α α · ∇ l k ZK = φ (ξ)u (ξ) ϕ (ξ)ϕ (ξ) J dξ α α · ∇ l k | | ZE i 1 im ∂ϕ˜l(ξ) = φ (ξ)u (ξ)(J− ) ϕ˜ (ξ) J dξ. α α ∂ξm k | | ZE In practise this integration must be done numerically. Gauss numerical integration formulae are used. The order of the integration is naturally kept the same as the interpolation order of the unknowns.

ξ2+1

ξ1 ξ x(ξ) → 1 +1 − x2

1 − x1

Figure 2.4: Mapping of the local reference element in ξ -coordinates to global element in physical x -coordinates for four point quadrilateral element

2.3.3 Solution of the discretized system The system of nonlinear equations (2.109) is transferred to algebraic system using the integration formula given above. To solve this resulting nonlinear system of algebraic equations a variety of methods exists. Nonlinear- ity results to iterative methods. The most naive approach would be to use fixed point iteration (Picard iteration), where the nonlinearity is eliminated taking the coefficient values from the previous iteration step. Unfor- tunately this method results to slow convergence. More effective way is to

91 2. Numerical methods use Newton-type methods, where the system is expanded using Taylor series in the neigborhood of the solution. This results to method with asymptot- ically quadratic convergence. Unfortunately the radius of convergence of this method can sometimes be quite small and care must be exercised in the choice of the initial solution vector. Convergence radius can be expanded using for example the so called incremental method where the right side load is incremented step by step. Another way is to take a few Picard iteration steps before proceeding to the Newton steps. We might also have to control the magnitude of the Newton step. This is needed when the Jacobian is not calculated exactly like in the Modified Newton-Raphson method, where the Jacobian is calulated only once in the beginning of the iterations or like in the Quasi-Newton method where the Jacobian is updated in a more simple manner than calculating it exactly or like in the inexact Newton methods where the Jacobian is evaluated from the residual vector using difference approximations. An effective way is to use line-search backtracking. For every nonlinear step we have to solve a linear system. When large systems are considered, like the system (2.109) mostly is, we have to use iterative solvers. Iterative solvers start with an initial guess and compute a sequence of approximate solutions that converge to the exact solution. The accuracy of the solutions depend on the number of iterations performed. When the linear equation solver is part of the nonlinear iteration loop, like we have, exact convergence is not required. The amount of work used with iterative solvers depends on the convergence rate and the desired accuracy. Depending on the system to be solved the convergence of the iterative methods can be slow and irregular. The rate of the convergence depends on the spectrum of the coefficient matrix and hence the condition number of the matrix. The rate of the convergence can often be increased by the technique of preconditioning. When the system of linear equations is nonsymmetric, like system (2.109), special methods must be used. Methods like Conjugate Gradient Squared (CGS) [SWd85] and the Generalized Minimum Resid- ual Method (GMRES) [SS83] are succesively used. For the solution of the (2.109) the GMRES method is succesfully used [Hil97]. In the following the Newton-GMRES algorithm with the linesearch backtracking is introduced. Let the nonlinear system of equations be R(U) = A(U)U F(U) = 0. Then the method as expressed in algorithmic form is − as follows,

Algorithm 2

U given, n = 1 0 − REPEAT n = n + 1 Solve J(Un)δ(n) = R(Un) by GMRES − Un+1 = Un + ωnδ(n)

92 2. Numerical methods

UNTIL convergence Above ωn is computed by line-search backtracking to decrease residual norm 1 t r(U) = 2 R(U) R(U).

Above the superscript n denotes the nth iteration step, i.e., previous iteration step. Matrix J(Un) is the Jacobian associated with R(Un), i.e.,

ik n i ij ij ∂A (U ) n ∂F J (n) = A + Uk + , (2.116) ∂Uj ∂Uj

th where Uj is the j component of the vector U in (2.109). Furthermore δ(n) is the desired search direction, ωn is the length of the step and R(Un) is the residual from the previous step. As a preconditioner for the GMRES the ILU preconditioning is preferred, which seems to work fairly well in the present problem of twoﬂuid ﬂow. For detailed analysis of Newton-GMRES algorithms (including also inexact Newton algorithms), see Ref. [CE93].

2.4 Particle tracking

From the algorithmic point of view the particle tracking is a considerably simple method as compared to multifluid algorithms. For the solution of the carrier fluid with low particle volume fractions standard one-phase algorithms can be used with extra source terms emanating from the appearance of the particulate phase. Particle trajectories are given by the kinematic equation. The treatment of this equation involves only technical difficulties. Equations of motion are reduced to ordinary differential equations and are therefore easy to solve. The coupling between the carrier phase and the particulate phase is taken into account through source terms. These terms can be obtained elementwise (or cellwise) keeping track of trajectories passing through the considered element. The solution strategy used in the particle tracking is an iterative one. First the solution of the carrier phase is obtained. Next the ordinary differential equations for the dispersed phase are solved for a number of particles using the continuum phase solution. The computed trajectories (and also other scalar quantities such as temperature and masses of particles) are combined into source terms of momentum equations (and of energy and continuity equations). These source terms are then used in the next solution of the continuum equations. The process is iterated until convergence is attained. In this procedure, two-way coupling is said to prevail since information is transfered from the carrier phase to dispersed phase and vice versa. When the particles are not affecting to the carrier phase and the tracking is such a postprocessing after the flow field of the carrier phase is solved one speaks of one-way coupling. The principle of numerical solution

93 2. Numerical methods

Start Start

Solution of fluid Solution of fluid phase flow field phase flow field without coupling to dispersed phase

Equation set converged No Equation set Yes No converged 1 Yes Lagrangian tracking of dispersed phase Solution of fluid entities Lagrangian tracking of dispersed phase phase flow field with entities phase coupling terms

Statistics adequate No Equation set Statistics adequate No No converged Yes Yes Yes End 1 End

Figure 2.5: Solution schemes for problems corresponding to one-way and two-way coupling. for one-way coupled systems and for two-way coupled systems is illustrated in Fig. 2.5. In this section the numerical realization of the particle tracking is shortly reviewed. All numerous technical details are excluded as well as special considerations for turbulent ﬂows. In turbulent tracking we refer here to the widely used work by Gosman and Ioannides [GI81b], where k ε turbulence − model for the carrier ﬂuid and a one-step-per-eddy, Monte-Carlo simulation for the particles were used.

2.4.1 Solution of the system of equation of motion

To solve the momentum equation (1.109) several numerical methods for ODE can be used. Let us write equation (1.109) in a form

d V = F(V, t). (2.117) dt

94 2. Numerical methods

In order to solve this system, an explicit second-order predictor-corrector method could be used, i.e., 1 1 Vn+1 = Vn + ∆t F(Vn+ 2 , tn+ 2 ) (2.118) 1 n+ 2 n 1 n n V = V + 2 ∆t F(V , t ). It is well known however that explicit methods perform poorly when system of equations becomes stiﬀ, i.e., when relaxation times becomes small. In such cases unconditionally stable implicit methods are preferred. An example is the following Crank-Nicholson second order scheme F(Vn, tn) + F(Vn+1, tn+1) Vn+1 = V n + ∆t . (2.119) 2 2.4.2 Solution of particle trajectories Particle trajectories are given by the kinematic equation d Y = V, (2.120) dt where Y(t) is the position of the particle at time t and V(t) the corresponding velocity. Using the ﬁnite element method for the equation (2.120) we get componentwise d Y i = V i(t)φ (ξj). (2.121) dt k k Xk The relationship between local coordinates ξj and the global coordinates Y i is given by the isoparametric mapping (2.113). Now we get for the reference element d ∂Y i d Y i = ξj = V i, (2.122) dt ∂ξj dt or in matrix form d d Y = J ξ = V, (2.123) dt dt where J is the Jacobian of the parametric transformation. So we can transform the original problem to a problem in reference element,

d 1 ξ = J− V. (2.124) dt Furthermore using second order Crank-Nicholson scheme for time derivative we get (J 1)jkV k(t ) + (J 1)jkV k(t ) ξj(t ) = ξj(t ) + ∆t − n − n+1 . (2.125) n+1 n 2 For ﬁnite volume methods the calculation of the trajectories is performed more or less the same way as within ﬁnite element method. Like in FEM, instead of dealing with physical coordinates computational coordinates are used.

95 2. Numerical methods

2.4.3 Source term calculation In the PSIC approach [CSS77], the source terms are calulated from the residence time of the each individual particle on each cell or element. j Let np be the number of particles per unit time traversing the jth trajec- j tory and δtE be the residence time of a particle on trajectory j with respect to element E. Then the contribution to the ith momentum transfer source to element E is

nE i 1 j j i j i S (E) = np (mpV )out (mpV )in , (2.126) VE − Xj=1 h i where nE is the number of trajectories passing through element E and VE is j i the volume of that element and (mpV )out is the momentum of the particle j i leaving the element and (mpV )in the momentum of the particle entering the element E with respect to the trajectory j.

2.4.4 Boundary condition When a particle reaches the domain boundary, the distribution of source terms between the carrier phase and the boundary depends on whether the particle penetrates, rebounds or attaches the boundary. A penetrating particle exchanges momentum (and, in general, energy and mass) with the carrier phase up to the boundary and then exits the domain carrying a residual of momentum (and of energy and mass). If the particle rebounds from the boundary, it exchanges momentum with the carrier phase and with the boundary face. If the particle attaches the boundary, it simply loses all its momemtum to the boundary.

2.5 Mesoscopic simulation methods

Numerical solution of flow has traditionally been based on finding a solution for partial differential continuum equations such as continuity and Navier- Stokes equations that govern the fluid flow. In principle, it would be possible to solve any fluid flow problem on microscopic level with direct molecular- dynamical simulations. It is quite clear that practical fluid flow problems cannot be solved with this approach at the present due to the large number of particles, and thus large amount of computer resources that would be needed for such a simulation. Another possibility for direct microscopic simulation of flow would be using the standard kinetic theory. Here the basic object is the particle distribution function f(r, v) which gives the probability of finding at position r a particle with velocity v. The time evolution of this function is given by the Boltzmann equation

F 3 0 0 (∂ +v r + v )f = dΩ d v δ(Ω) v v (f f f f ). (2.127) t 1 ·∇ m ·∇ 1 1 2 | 1 − 2| 2 1 − 1 2 Z Z 96 2. Numerical methods

This approach is also far too complicated in most practical problems even for gaseous fluids. Many dynamical systems can however be modeled with radically simplified microscopic or mesosocopic models. This has been utilized, e.g., in the famous Ising models for magnetic materials, and since 1950’s in various cellular-automaton models for biological and physical systems. Encouraged by these models, similar models (i.e. models that were not based on continuum mechanics) for fluid flows were also developed, first with a limited success, though. In 1986 it was realised that fluid flow could be successfully simulated with very simple discrete models provided that the simulation lattice was carefully chosen. This discovery was the starting point of the lattice-gas methods [FdH+87, RZ97, Kop98] and later of the lattice-Boltzmann methods [RZ97, Kop98, QdL92, BSV92]. We refer these methods as ’mesoscopic’ indicating that the size scale of the basic constituents (’particles’) or variables of the model is large as compared to molecular scale of the fluid, but small as compared to the typical size scale of macroscopic flow, and the local hydrodynamical quantities are defined as suitable averages of the basic variables. In the lattice-gas method fluid is modeled with identical particles which move in a discrete lattice interacting with each other only at the lattice nodes. A ’particle’ is not considered here as a physical molecule, but rather as a ’fluid particle’ that contains a large number of molecules. The space, the time and the velocity of particles are all discrete. Notice that the method is not based on explicitly solving any governing equation of particle motion. It is simply an algorithm of moving particles in the lattice with a set of collision rules that conserve mass (number of particles) and momentum. The basic hydrodynamic quantities such as local flow velocity can be defined as the velocity of particles averaged over a specified volume, time or both. Provided that the simulation lattice fulfills certain symmetry requirements, the lattice-gas automaton indeed develops macroscopic flow field that is close to the solution of the Navier-Stokes equations for incompressible flow in the same macroscopic conditions. The number of particles needed for a realistic simulation of many practical systems is well in the limits of present computational capabilities. The boolean and local nature of the lattice-gas model gives several technical advantages in computing. Perhaps the most important advantage is the ease of introducing complex flow geometries, which follows from the regular lattice and from the extremely simple and strictly local updating rules of the method. Since particles can be represented by single bits in computer, the method does not require very large memory as compared to other methods. Furthermore, rounding errors are not involved in bit manipulations, and unconditional numerical stability is guaranteed. The computations are also inherently parallel thus being ideal for massive parallel computers. In recent years many sophistications have helped to get rid of most of the early problems [RZ97, BSV92], and the lattice-gas method is now a viable tool

97 2. Numerical methods for computational fluid dynamics. The early deficiencies of the lattice-gas models inspired the formulation of the more advanced lattice-Boltzmann models. The idea behind this model was to track a population of (fluid) particles instead of a single particle, a reasonable modification justified by the Boltzmann molecular-chaos assumption from kinetic theory of gases. This mean-value representation of particles eliminates much of the statistical noise present in lattice-gas methods. The basic variable to be solved within lattice-Boltzmann methods is the discrete distribution function that fulfils a discretised version of the Boltzmann equation (see eqn. (2.128) below). The hydrodynamic variables are defined in terms of the distribution function as averaged quantities analogously with the manner in which those variables are defined in the usual kinetic theory of gases. In the next section we discuss in more detail one of the simplest versions of the lattice-Boltzmann models that is regularly used in practical simulations, namely the lattice-BGK (Bhatnagar-Gross-Krook) model [QdL92]. Recently a third mesoscopic method, dissipative particle dynamics (DPD), has been developed [MBE97]. This method is a combination of molecular dynamics, Brownian dynamics and lattice-gas automata. The DPD algorithm models a fluid with fluid particles out of equilibrium and conserves mass and momentum. The fluid particles interact with each other through conservative and dissipative forces. In contrast to the lattice-gas and lattice- Boltzmann methods, the position and velocity of these fluid particles are continuous, as in molecular dynamics, but time is discrete. This method is very suitable e.g. for simulating rheological properties of complex fluids on hydrodynamic time scales.

2.5.1 The lattice-BGK model In the lattice-Boltzmann method a simpliﬁed version of the Boltzmann equation, eqn. (2.127), is solved on a discrete, regular lattice. Fluid particles, described by the discrete distribution function, move synchronously along the bonds of this lattice, and interact locally according to a given set of rules. A single iteration step (time step) within this method consists of the following two phases:

1. Propagation; particles move along lattice bonds to the neighboring lattice nodes.

2. Collision; particles on the same lattice node shuﬄe their velocities locally conserving mass and momentum.

In the lattice-BGK (Bhatnagar-Gross-Krook) model the collision operator is based on a single time relaxation to the local equilibrium distribution [QdL92]. The lattice structure can be chosen in several ways (see Fig. 2.6

98 2. Numerical methods for two possible realizations). In the D3Q19 lattice-BGK model, each lattice point is linked in three dimensional space with its six nearest neighbors at a unit distance and with twelve diagonal neighbors at a distance of √2. Including the rest state, the particles can thus have 19 diﬀerent velocity states. The dynamics of the D3Q19 model is given by the equation [RZ97, QdL92], 1 f (r + c , t + 1) = f (r, t) + (f (0)(r, t) f (r, t)) , (2.128) i i i τ i − i where ci is the i-th link, fi(r, t) is the density of particles moving in the (0) ci-direction, τ is the BGK relaxation parameter, and fi (r, t) is the equilibrium distribution function towards which the particle populations are relaxed. The hydrodynamic ﬁelds such as the density ρ, the velocity v and the momentum tensor Π are obtained from moments of the discrete velocity distribution fi(r, t) as

18 18 f (r, t)c ρ(r, t) = f (r, t), v(r, t) = i=0 i i , i ρt (r, ) Xi=0 P 18 Παβ(r, t) = ciαciβfi(r, t). Xi=0 The equilibrium distribution function can be chosen in several ways. A common choice is

(0) 1 1 2 1 2 fi = tiρ(1 + 2 (ci v) + 4 (ci v) 2 v ), cs · 2cs · − 2cs where ti is a weight factor depending on the length of the link vector and 1 1 cs is the speed of sound. The weight factors can be chosen as 3 , 18 and 1 36 for the rest particle, nearest neighboring and diagonal neighboring links, respectively. These values yield to a correct hydrodynamic behavior for an incompressible ﬂuid in the limit of low Mach and Knudsen numbers. The speed of sound and the kinematic viscosity of the simulated ﬂuid in lattice units are given by c = 1 and ν = 2τ 1 , respectively [QdL92]. s √3 6−

2.5.2 Boundary conditions In lattice-gas and lattice-Boltzmann simulations the no-slip boundary condition is usually realized using the bounce-back condition [RZ97, Kop98]. In this approach the momenta of particles that meet wall points are simply re- versed. The bounce-back boundary may generate errors which in some cases violate the second-order spatial convergence of these methods and more so- phisticated boundaries have been proposed [FH97]. For practical simulations the bounce-back boundary is however very attractive, because it is a simple

99 2. Numerical methods

Figure 2.6: Two possible realizations of lattice structures. and computationally efficient method for imposing no-slip walls in irregular geometries. Also, the bounce-back method can easily be generalized to allow moving boundaries [Lad94a, Lad94b]. Successful numerical simulation of practical fluid flow problems requires that the velocity and pressure boundary conditions of the system have been imposed in a consistent way. So far most of the practical simulations have used a body force [Kop98] instead of pressure or velocity boundaries. When a body force is used, a pressure gradient acting on the fluid is replaced by a uniform external force. (Usually periodic boundaries are imposed at least in the direction of the flow.) The use of a body force is based on the assumption that the effect of an external pressure gradient is approximately constant all over the system, and that it can be replaced by a constant force that adds at every time step a fixed amount of momentum to fluid particles. In a simple tube flow the body-force approach is exact. In more complicated geometries this approach is supposed to work best with small Reynolds numbers where nonlinear effects on the flow are small.

2.5.3 Liquid-particle suspensions The bounce-back condition can easily be modified [Lad94a] to allow moving boundaries with no-slip boundary condition. We assume that a boundary moving with velocity uw is located in the halfway of links between the last fluid points and the first solid points. The distribution function for particles moving along such a link towards the solid point is given by

f (r + c , t + 1) = f 0 (r + c , t ) + 2ρ B (u c ). (2.129) i i i i + f i w · i Here t+ is used to indicate the post-collision distribution, i0 denotes the 1 1 bounce-back link, and Bi = 3 , 12 for the diagonal and non-diagonal links, respectively. The last term in Eqn. 2.129 is added in order to account for the momentum transfer between the fluid and the moving solid boundary. For computational convenience the fluid is usually made to fill also the suspended solid particles. This trick removes the need for creating and destroying fluid when particles move. Models without interior fluid have also been used.

100 2. Numerical methods

The lattice-Boltzmann method for suspensions is based upon Newtonian dynamics of solid particles that move in a continuous space. The discretized images of these particles interact with the lattice-Boltzmann fluid at their boundary nodes. The technique takes advantage of the fact that the hydrodynamic interactions are time dependent and develop from purely local interactions at the solid-liquid interfaces. Thus it is not necessary to consider the global system, but one can update one particle at a time. The method scales linearly with the number of suspended particles and, therefore, allows far larger simulations than the conventional methods. The hydrodynamic interactions between solid particles are fully accounted for, both at zero and finite Reynolds numbers [Lad94a]. Furthermore, there is no need for solution of the linear systems. Therefore, lattice-Boltzmann method can be efficiently implemented on parallel processors. The electrostatic interactions, the flow geometry, the Peclet number, the shear and particle Reynolds numbers, as well as the size and shape of the suspended particles can easily be varied.

2.5.4 Applicability of mesoscopic methods

The most important property of the mesoscopic methods discussed above is the simplicity with which models for many complex flows such as multiphase flows and flows in porous media can be implemented. Numerical simulation is based on a simple algorithm, or on numerical solution of a single transport equation instead of a system of coupled partial differential equations of conventional methods. Another important property of the lattice-gas and lattice-Boltzmann methods is the inherent spatial locality of their updating rules, which makes these methods ideal for parallel processing. Both methods are also numerically relatively stable. In the lattice-gas and lattice- Boltzmann methods a uniform lattice is usually used, and they can thus be easily and quickly applied to new geometries. Also, time-dependent simulations can be carried out relatively easily, as no time is lost for remeshing. Moreover, these methods spontaneously generate hydrodynamic instabili- ties which make them useful for simulating fluid flow at moderately large Reynolds numbers. At very large Reynolds numbers mesoscopic methods encounter the problem of turbulence like the more conventional methods: in a fully turbulent flow wave lengths of all scales are present, and the solution would require a very large lattice or additional turbulence modelling. Furthermore, some features of the methods such as use of irregular lattice for improved accuracy, implementation of velocity and pressure boundary conditions and models for finite-temperature systems are still under development. So far, successful implementations of mesoscopic methods have included e.g. multiphase flows [RZ97, MC96], suspension flows [Lad94a] and flows in complex geometries [KKHA98, Kop98, MC96]. In Ref. [KVH+99] a

101 2. Numerical methods

Figure 2.7: Couette flow of liquid-particle suspension as solved using the lattice-BGK-method in two dimensions. The upper wall is moving to right and the lower wall is fixed. The velocity of the liquid is indicated by gray scale (light colour indicating high velocity) and the motion and rotation of particles by arrows. detailed comparison between conventional methods and lattice-Boltzmann methods was given for 3D fluid flow in an industrial static mixer. In addition to basic flow simulation, these methods are convenient in finding numerical correlations for closure relations and constitutive relations necessary in conventional methods for solving multiphase flows, as discussed in chapter 1. The essential property of the mesoscopic methods in this respect is the simplicity of constructing complicated geometries and implementing moving boundaries. Due to this feature it is straightforward, e.g., to generate a large ensemble of macroscopically identical systems of particles suspended in a liquid, solve the flow and the motion of the particles for each system (possibly with a given interaction between the suspended particles), and finally compute the properly averaged quantities and transfer integrals as defined in chapter 1. The results can then be used to correlate the unknown terms such as Γα and Mα in averaged flow equations (see eqns. (1.27)–(1.28) or eqns. (1.79)–(1.80)) with basic averaged flow quantities. Similarly, these methods could be helpful in finding rheological properties and boundary conditions of multiphase fluids to be used in conventional non-Newtonian simulation of such flows. As an example of a typical solution obtained by the lattice-Boltzmann method, we show in Figure 2.7 the instantaneous flow pattern of a two- dimensional liquid-particle suspension in a shear flow [RSMK+00]. Here, the flow of the carrier fluid is calculated using the lattice-BGK -model while the motion of the suspended spherical particles is given by Newtonian mechanics including the impulsive forces due to direct collisions between particles, and the hydrodynamic forces due to fluid flow. The numerical solution can be

102 2. Numerical methods

Figure 2.8: Viscosity of two-dimensional liquid-particle suspension as a function of consistency (area fraction) calculated by lattice-Boltzmann simulations (see Fig. 2.7). The open squares are the calculated values. The semiempirical result by Krieger and Dougherty [KD59], and the analytical results by Batchelor [BG72, Bat77] and Einstein [Ein06, Ein11] valid for low consistencies are given by lines as indicated. used to compute, e.g., the time averaged total shear force acting on the walls, and thereby to ﬁnd the dependence of the nominal viscosity on the mean shear velocity and on the properties of the suspension. Figure 2.8 shows the viscosity of the two dimensional liquid-particle suspension as a function of consistency of particles as calculated using lattice-Boltzmann simulations. The result is also compared with previous analytical results by Einstein [Ein06, Ein11] and Batchelor [BG72, Bat77], and with the semiempirical results by Krieger and Dougherty [KD59].

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113 Appendix 1

Submatrices and vectors for the ﬁnite-element matrix system (2.109):

11 11 22 33 Akl = 2Kf (k, l) + Kf (k, l) + Kf (k, l) + Cf (k, l) + If(k, l) c I M 11 +Sf (k, l) + Sf (k, l) + Sf (k, l) 12 12 M 12 Akl = Kf (k, l) + Sf (k, l) 13 13 M 13 Akl = Kf (k, l) + Sf (k, l) 14 1 p1 Akl = Pf (k, l) + Sf (k, l) 15 − I Akl = If(k, l) Sf (k, l) 1i − − Akl = 0, i = 6, 7, 8 21 21 M 21 Akl = Kf (k, l) + Sf (k, l) 22 11 22 33 Akl = Kf (k, l) + 2Kf (k, l) + Kf (k, l) + Cf (k, l) + If(k, l) c I M 22 +Sf (k, l) + Sf (k, l) + Sf (k, l) 23 23 M 23 Akl = Kf (k, l) + Sf (k, l) 24 2 p2 Akl = Pf (k, l) + Sf (k, l) 25 − Akl = 0 26 I Akl = If(k, l) Sf (k, l) 2i − − Akl = 0, i = 7, 8 31 31 M 31 Akl = Kf (k, l) + Sf (k, l) 32 32 M 32 Akl = Kf (k, l) + Sf (k, l) 33 11 22 33 Akl = Kf (k, l) + Kf (k, l) + 2Kf (k, l) + Cf (k, l) + If(k, l) c I M 33 +Sf (k, l) + Sf (k, l) + Sf (k, l) 34 3 p3 Akl = Pf (k, l) + Sf (k, l) 3i − Akl = 0, i = 5, 6 37 I Akl = If(k, l) Sf (k, l) 38 − − Akl = 0 1 1 A41 = P 1(l, k) Sqc (k, l) SqI kl − f − f − f 1/1 Appendix 1

2 2 42 2 qc qI Akl = Pf (l, k) Sf (k, l) Sf (k, l) − − 3 − 3 A43 = P 3(l, k) Sqc (k, l) SqI (k, l) kl − f − f − f 44 qp Akl = Sf (k, l) − 1 45 qI Akl = Sf (k, l) 2 46 qI Akl = Sf (k, l) 3 47 qI Akl = Sf (k, l) 48 Akl = 0 51 I Akl = IA(k, l) SA(k, l) 5i − − Akl = 0, i = 2, 3 54 1 p1 Akl = PA(k, l) + SA (k, l) 55 − 11 22 33 Akl = 2KA (k, l) + KA (k, l) + KA (k, l) + CA(k, l) + IA(k, l) c I +SA(k, l) + SA(k, l) 56 12 Akl = KA (k, l) 57 13 Akl = KA (k, l) 58 Akl = 0 61 Akl = 0 62 I Akl = IA(k, l) SA(k, l) 63 − − Akl = 0 64 2 p2 Akl = PA(k, l) + SA (k, l) 65 − 21 Akl = KA (k, l) 66 11 22 33 Akl = KA (k, l) + 2KA (k, l) + KA (k, l) + CA(k, l) + IA(k, l) c I +SA(k, l) + SA(k, l) 67 23 Akl = KA (k, l) 68 Akl = 0 71 Akl = 0 72 Akl = 0 A73 = I (k, l) SI (k, l) kl − A − A 74 3 p3 Akl = PA(k, l) + SA (k, l) 75 − 31 Akl = KA (k, l) 76 32 Akl = KA (k, l) 77 11 22 33 Akl = KA (k, l) + KA (k, l) + 2KA (k, l) + CA(k, l) + IA(k, l) c I +SA(k, l) + SA(k, l) 78 Akl = 0 1 81 qI Akl = SA (k, l)

1/2 Appendix 1

2 82 qI Akl = SA (k, l) 3 83 qI Akl = SA (k, l) 84 qp Akl = SA (k, l) − 1 1 85 qc qI Akl = SA (k, l) SA (k, l) − 2 − 2 86 qc qI Akl = SA (k, l) SA (k, l) − 3 − 3 A87 = Sqc (k, l) SqI (k, l) kl − A − A A88 = DM (k, l) P¯1 (l, k) P¯2 (l, k) P¯3 (l, k) kl − − A − A − A ¯M 11 ¯M 12 ¯M 13 +SA (k, l) + SA (k, l) + SA (k, l) ¯M 21 ¯M 22 ¯M 23 +SA (k, l) + SA (k, l) + SA (k, l) ¯M 31 ¯M 32 ¯M 33 +SA (k, l) + SA (k, l) + SA (k, l),

where the indices get values k, l = 1, . . . , ndofs. The integral forms above are deﬁned as

ij ∂ϕl ∂ϕk Dα (k, l) = (1/Reα)(φα , ) ∂xi ∂xj C (k, l) = (φ u ϕ , ϕ ) α α α · ∇ l k i ∂ϕk ∂φα Pα(k, l) = (ϕl, βα(φα + ϕk )) ∂xi ∂xi ¯i ∂uαi ∂ϕk Pα(k, l) = (ϕl, βα(ϕk + uαi )) ∂xi ∂xi Iα(k, l) = (gβαϕl, ϕk) Sc (k, l) = (φ u ϕ , τ u ϕ ) α α α · ∇ l α α · ∇ k pi ∂ϕl Sα (k, l) = (βαφα , ταuα ϕk) ∂xi · ∇ SI (k, l) = (gβ ϕ , τ u ϕ ) α α l α α · ∇ k i qc ∂ϕk Sα (k, l) = (φαuα ϕl, ταβα ) · ∇ ∂xi qp Sα (k, l) = (β φ ϕ , τ β ϕ ) α α∇ l α α∇ k i qI ∂ϕk Sα (k, l) = (gβαϕl, ταβα ) ∂xi M ij ∂ϕl ∂φα ∂ϕk ∂φα Sα (k, l) = (βα(φα + ϕl ), δαβα(φα + ϕk )) ∂xj ∂xj ∂xi ∂xi ¯M ij ∂uαj ∂ϕl ∂uαi ∂ϕk Sα (k, l) = (βα(ϕl + uαj ), δαβα(ϕk + uαi )) ∂xj ∂xj ∂xi ∂xi DM (k, l) = (β κ ϕ , ϕ ). A ∇ l ∇ k

1/3 Appendix 1

The subvectors of the right hand side vector F are deﬁned as

F (k) = (φ β F˜i, ϕ + τ (φ u ϕ )), i = 1, 2, 3 i f f f k f f f · ∇ k F4(k) = (φf βf F˜ f , τf φf βf ϕk) i−4 ∇ F (k) = (φ β F˜ − , ϕ + τ (φ u ϕ )), i = 5, 6, 7 i A A A k A A A · ∇ k F (k) = (φ β F˜ , τ φ β ϕ ), 8 A A A − A A A∇ k where k = 1, . . . , ndofs.

1/4

Series title, number and report code of publication

VTT Publications 722 VTT-PUBS-722 Author(s) Kai Hiltunen, Ari Jäsberg, Sirpa Kallio, Hannu Karema, Markku Kataja, Antti Koponen, Mikko Manninen & Veikko Taivassalo Title Multiphase Flow Dynamics Theory and Numerics Abstract The purpose of this work is to review the present status of both theoretical and numerical research of multiphase flow dynamics and to make the results of that fundamental research more readily available for students and for those working with practical problems involving multiphase flow. Flows that appear in many of the common industrial processes are intrinsically multiphase flows. The advanced technology associated with these flows has great economical value. Nevertheless, our basic knowledge and understanding of these processes is often quite limited as, in general, is our capability of solving these flows. In the first part of this publication we give a comprehensive review of the theory of multiphase flows accounting for several alternative approaches. We also give general guidelines for solving the ’closure problem’, which involves, e.g., characterising the interactions between different phases and thereby deriving the final closed set of equations for the particular multiphase flow under consideration. The second part is devoted to numerical methods for solving those equations.

ISBN 978-951-38-7365-3 (soft back ed.) 978-951-38-7366-0 (URL: http://www.vtt.fi/publications/index.jsp)

Series title and ISSN Project number VTT Publications 33861 1235-0621 (soft back ed.) 1455-0849 (URL: http://www.vtt.fi/publications/index.jsp)

Date Language Pages December 2009 English, Finnish abstr. 113 p. + app. 4 p.

Name of project Commissioned by Dynamics of Multiphase Flows, Finnish national Computational Fluid Dynamics Technology Programme 1995–1999

Keywords Publisher Multiphase flows, volume averaging, ensemble VTT Technical Research Centre of Finland averaging, mixture models, multifluid finite P.O. Box 1000, FI-02044 VTT, Finland volume method, multifluid finite element method, Phone internat. +358 20 722 4520 particle tracking, the lattice-BGK model Fax +358 20 722 4374

Julkaisun sarja, numero ja raporttikoodi VTT Publications 722 VTT-PUBS-722

Tekijä(t) Kai Hiltunen, Ari Jäsberg, Sirpa Kallio, Hannu Karema, Markku Kataja, Antti Koponen, Mikko Manninen & Veikko Taivassalo Nimeke Monifaasivirtausten dynamiikka Teoriaa ja numeriikkaa Tiivistelmä Työssä tarkastellaan monifaasivirtausten teoreettisen ja numeerisen tutkimuksen nykytilaa, ja muodostetaan tuon perustutkimuksen tuloksista selkeä kokonaisuus opiskelijoiden ja käytännön virtausongelmien kanssa työskentelevien käyttöön. Monissa teollisissa prosesseissa esiintyvät virtaukset ovat olennaisesti monifaasivirtauksia – esimerkiksi kaasu-partikkeli-, neste-partikkeli- ja neste-kuitu- suspensioiden virtaukset, sekä kuplavirtaukset, neste-neste-virtaukset ja virtaus huokoisen aineen läpi. Julkaisun ensimmäisessä osassa tarkastellaan kattavasti monifaasivirtausten teoriaa ja esitetään useita vaihtoehtoisia lähestymistapoja. Toisessa osassa käydään läpi monifaasivirtauksia kuvaavien yhtälöiden numeeri- sia ratkaisumenetelmiä.

ISBN 978-951-38-7365-3 (nid.) 978-951-38-7366-0 (URL: http://www.vtt.fi/publications/index.jsp)

Avainnimeke ja ISSN Projektinumero VTT Publications 33861 1235-0621 (nid.) 1455-0849 (URL: http://www.vtt.fi/publications/index.jsp)

Julkaisuaika Kieli Sivuja Joulukuu 2009 Englanti, suom. tiiv. 113 s. + liitt. 4 s.

Projektin nimi Toimeksiantaja(t) Dynamics of Multiphase Flows, Finnish national Computational Fluid Dynamics Technology Programme 1995–1999 Avainsanat Julkaisija Multiphase flows, volume averaging, ensemble VTT averaging, mixture models, multifluid finite PL 1000, 02044 VTT volume method, multifluid finite element method, Puh. 020 722 4520 particle tracking, the lattice-BGK model Faksi 020 722 4374

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722

The purpose of this work is to review the present status of both theoretical and M numerical research of multiphase flow dynamics and to make the results of that ultip fundamental research more readily available for students and for those working with h practical problems involving multiphase flow. as e

Flows that appear in many of the common industrial processes are intrinsically F multiphase flows. The advanced technology associated with these flows has great lo economical value. Nevertheless, our basic knowledge and understanding of these w processes is often quite limited as, in general, is our capability of solving these flows. Dy namics In the first part of this publication we give a comprehensive review of the theory of multiphase flows accounting for several alternative approaches. We also give general .

guidelines for solving the ’closure problem’, which involves, e.g., characterising the interactions between different phases and thereby deriving the final closed set of The equations for the particular multiphase flow under consideration. The second part or y is devoted to numerical methods for solving those equations. and N um e rics

ISBN 978-951-38-7365-3 (soft back ed.) ISBN 978-951-38-7366-0 (URL: http://www.vtt.fi/publications/index.jsp) ISSN 1235-0621 (soft back ed.) ISSN 1455-0849 (URL: http://www.vtt.fi/publications/index.jsp)